The short answer:
The difference is minimal. Code readability is probably your biggest gain, depending on your needs. There are tradeoffs with both approaches.
The long answer:
I dumped both variants into Sharplab.io to see what code compilation would look like.
Variant 1:
using System;
using System.Threading.Tasks;
public class C {
    public async void M() {
        MyFunction(await GetObject1(), await GetObject2());
    }
    public async Task<Object> GetObject1(){
        return new object();
    }
    public async Task<Object> GetObject2(){
        return new object();
    }
    public void MyFunction(Object object1, Object object2) {
    
    }
}
compiles to...
using System;
using System.Diagnostics;
using System.Reflection;
using System.Runtime.CompilerServices;
using System.Runtime.InteropServices;
using System.Security;
using System.Security.Permissions;
using System.Threading.Tasks;
[assembly: CompilationRelaxations(8)]
[assembly: RuntimeCompatibility(WrapNonExceptionThrows = true)]
[assembly: Debuggable(DebuggableAttribute.DebuggingModes.IgnoreSymbolStoreSequencePoints)]
[assembly: SecurityPermission(SecurityAction.RequestMinimum, SkipVerification = true)]
[assembly: AssemblyVersion("0.0.0.0")]
[module: UnverifiableCode]
public class C
{
    [StructLayout(LayoutKind.Auto)]
    [CompilerGenerated]
    private struct <M>d__0 : IAsyncStateMachine
    {
        public int <>1__state;
        public AsyncVoidMethodBuilder <>t__builder;
        public C <>4__this;
        private object <>7__wrap1;
        private TaskAwaiter<object> <>u__1;
        private void MoveNext()
        {
            int num = <>1__state;
            C c = <>4__this;
            try
            {
                TaskAwaiter<object> awaiter;
                if (num != 0)
                {
                    if (num == 1)
                    {
                        awaiter = <>u__1;
                        <>u__1 = default(TaskAwaiter<object>);
                        num = (<>1__state = -1);
                        goto IL_00ca;
                    }
                    awaiter = c.GetObject1().GetAwaiter();
                    if (!awaiter.IsCompleted)
                    {
                        num = (<>1__state = 0);
                        <>u__1 = awaiter;
                        <>t__builder.AwaitUnsafeOnCompleted(ref awaiter, ref this);
                        return;
                    }
                }
                else
                {
                    awaiter = <>u__1;
                    <>u__1 = default(TaskAwaiter<object>);
                    num = (<>1__state = -1);
                }
                <>7__wrap1 = awaiter.GetResult();
                awaiter = c.GetObject2().GetAwaiter();
                if (!awaiter.IsCompleted)
                {
                    num = (<>1__state = 1);
                    <>u__1 = awaiter;
                    <>t__builder.AwaitUnsafeOnCompleted(ref awaiter, ref this);
                    return;
                }
                goto IL_00ca;
                IL_00ca:
                object result = awaiter.GetResult();
                c.MyFunction(<>7__wrap1, result);
                <>7__wrap1 = null;
            }
            catch (Exception exception)
            {
                <>1__state = -2;
                <>t__builder.SetException(exception);
                return;
            }
            <>1__state = -2;
            <>t__builder.SetResult();
        }
        void IAsyncStateMachine.MoveNext()
        {
            //ILSpy generated this explicit interface implementation from .override directive in MoveNext
            this.MoveNext();
        }
        [DebuggerHidden]
        private void SetStateMachine(IAsyncStateMachine stateMachine)
        {
            <>t__builder.SetStateMachine(stateMachine);
        }
        void IAsyncStateMachine.SetStateMachine(IAsyncStateMachine stateMachine)
        {
            //ILSpy generated this explicit interface implementation from .override directive in SetStateMachine
            this.SetStateMachine(stateMachine);
        }
    }
    [StructLayout(LayoutKind.Auto)]
    [CompilerGenerated]
    private struct <GetObject1>d__1 : IAsyncStateMachine
    {
        public int <>1__state;
        public AsyncTaskMethodBuilder<object> <>t__builder;
        private void MoveNext()
        {
            object result;
            try
            {
                result = new object();
            }
            catch (Exception exception)
            {
                <>1__state = -2;
                <>t__builder.SetException(exception);
                return;
            }
            <>1__state = -2;
            <>t__builder.SetResult(result);
        }
        void IAsyncStateMachine.MoveNext()
        {
            //ILSpy generated this explicit interface implementation from .override directive in MoveNext
            this.MoveNext();
        }
        [DebuggerHidden]
        private void SetStateMachine(IAsyncStateMachine stateMachine)
        {
            <>t__builder.SetStateMachine(stateMachine);
        }
        void IAsyncStateMachine.SetStateMachine(IAsyncStateMachine stateMachine)
        {
            //ILSpy generated this explicit interface implementation from .override directive in SetStateMachine
            this.SetStateMachine(stateMachine);
        }
    }
    [StructLayout(LayoutKind.Auto)]
    [CompilerGenerated]
    private struct <GetObject2>d__2 : IAsyncStateMachine
    {
        public int <>1__state;
        public AsyncTaskMethodBuilder<object> <>t__builder;
        private void MoveNext()
        {
            object result;
            try
            {
                result = new object();
            }
            catch (Exception exception)
            {
                <>1__state = -2;
                <>t__builder.SetException(exception);
                return;
            }
            <>1__state = -2;
            <>t__builder.SetResult(result);
        }
        void IAsyncStateMachine.MoveNext()
        {
            //ILSpy generated this explicit interface implementation from .override directive in MoveNext
            this.MoveNext();
        }
        [DebuggerHidden]
        private void SetStateMachine(IAsyncStateMachine stateMachine)
        {
            <>t__builder.SetStateMachine(stateMachine);
        }
        void IAsyncStateMachine.SetStateMachine(IAsyncStateMachine stateMachine)
        {
            //ILSpy generated this explicit interface implementation from .override directive in SetStateMachine
            this.SetStateMachine(stateMachine);
        }
    }
    [AsyncStateMachine(typeof(<M>d__0))]
    public void M()
    {
        <M>d__0 stateMachine = default(<M>d__0);
        stateMachine.<>t__builder = AsyncVoidMethodBuilder.Create();
        stateMachine.<>4__this = this;
        stateMachine.<>1__state = -1;
        stateMachine.<>t__builder.Start(ref stateMachine);
    }
    [AsyncStateMachine(typeof(<GetObject1>d__1))]
    public Task<object> GetObject1()
    {
        <GetObject1>d__1 stateMachine = default(<GetObject1>d__1);
        stateMachine.<>t__builder = AsyncTaskMethodBuilder<object>.Create();
        stateMachine.<>1__state = -1;
        stateMachine.<>t__builder.Start(ref stateMachine);
        return stateMachine.<>t__builder.Task;
    }
    [AsyncStateMachine(typeof(<GetObject2>d__2))]
    public Task<object> GetObject2()
    {
        <GetObject2>d__2 stateMachine = default(<GetObject2>d__2);
        stateMachine.<>t__builder = AsyncTaskMethodBuilder<object>.Create();
        stateMachine.<>1__state = -1;
        stateMachine.<>t__builder.Start(ref stateMachine);
        return stateMachine.<>t__builder.Task;
    }
    public void MyFunction(object object1, object object2)
    {
    }
}
Variant 2:
using System;
using System.Threading.Tasks;
public class C {
    public async void M() {
        var obj1 = await GetObject1();
        var obj2 = await GetObject2();
        MyFunction(obj1, obj2);
    }
    public async Task<Object> GetObject1(){
        return new object();
    }
    public async Task<Object> GetObject2(){
        return new object();
    }
    public void MyFunction(Object object1, Object object2) {
    
    }
}
compiles to...
using System;
using System.Diagnostics;
using System.Reflection;
using System.Runtime.CompilerServices;
using System.Runtime.InteropServices;
using System.Security;
using System.Security.Permissions;
using System.Threading.Tasks;
[assembly: CompilationRelaxations(8)]
[assembly: RuntimeCompatibility(WrapNonExceptionThrows = true)]
[assembly: Debuggable(DebuggableAttribute.DebuggingModes.IgnoreSymbolStoreSequencePoints)]
[assembly: SecurityPermission(SecurityAction.RequestMinimum, SkipVerification = true)]
[assembly: AssemblyVersion("0.0.0.0")]
[module: UnverifiableCode]
public class C
{
    [StructLayout(LayoutKind.Auto)]
    [CompilerGenerated]
    private struct <M>d__0 : IAsyncStateMachine
    {
        public int <>1__state;
        public AsyncVoidMethodBuilder <>t__builder;
        public C <>4__this;
        private object <>7__wrap1;
        private TaskAwaiter<object> <>u__1;
        private void MoveNext()
        {
            int num = <>1__state;
            C c = <>4__this;
            try
            {
                TaskAwaiter<object> awaiter;
                if (num != 0)
                {
                    if (num == 1)
                    {
                        awaiter = <>u__1;
                        <>u__1 = default(TaskAwaiter<object>);
                        num = (<>1__state = -1);
                        goto IL_00ca;
                    }
                    awaiter = c.GetObject1().GetAwaiter();
                    if (!awaiter.IsCompleted)
                    {
                        num = (<>1__state = 0);
                        <>u__1 = awaiter;
                        <>t__builder.AwaitUnsafeOnCompleted(ref awaiter, ref this);
                        return;
                    }
                }
                else
                {
                    awaiter = <>u__1;
                    <>u__1 = default(TaskAwaiter<object>);
                    num = (<>1__state = -1);
                }
                <>7__wrap1 = awaiter.GetResult();
                awaiter = c.GetObject2().GetAwaiter();
                if (!awaiter.IsCompleted)
                {
                    num = (<>1__state = 1);
                    <>u__1 = awaiter;
                    <>t__builder.AwaitUnsafeOnCompleted(ref awaiter, ref this);
                    return;
                }
                goto IL_00ca;
                IL_00ca:
                object result = awaiter.GetResult();
                c.MyFunction(<>7__wrap1, result);
                <>7__wrap1 = null;
            }
            catch (Exception exception)
            {
                <>1__state = -2;
                <>t__builder.SetException(exception);
                return;
            }
            <>1__state = -2;
            <>t__builder.SetResult();
        }
        void IAsyncStateMachine.MoveNext()
        {
            //ILSpy generated this explicit interface implementation from .override directive in MoveNext
            this.MoveNext();
        }
        [DebuggerHidden]
        private void SetStateMachine(IAsyncStateMachine stateMachine)
        {
            <>t__builder.SetStateMachine(stateMachine);
        }
        void IAsyncStateMachine.SetStateMachine(IAsyncStateMachine stateMachine)
        {
            //ILSpy generated this explicit interface implementation from .override directive in SetStateMachine
            this.SetStateMachine(stateMachine);
        }
    }
    [StructLayout(LayoutKind.Auto)]
    [CompilerGenerated]
    private struct <GetObject1>d__1 : IAsyncStateMachine
    {
        public int <>1__state;
        public AsyncTaskMethodBuilder<object> <>t__builder;
        private void MoveNext()
        {
            object result;
            try
            {
                result = new object();
            }
            catch (Exception exception)
            {
                <>1__state = -2;
                <>t__builder.SetException(exception);
                return;
            }
            <>1__state = -2;
            <>t__builder.SetResult(result);
        }
        void IAsyncStateMachine.MoveNext()
        {
            //ILSpy generated this explicit interface implementation from .override directive in MoveNext
            this.MoveNext();
        }
        [DebuggerHidden]
        private void SetStateMachine(IAsyncStateMachine stateMachine)
        {
            <>t__builder.SetStateMachine(stateMachine);
        }
        void IAsyncStateMachine.SetStateMachine(IAsyncStateMachine stateMachine)
        {
            //ILSpy generated this explicit interface implementation from .override directive in SetStateMachine
            this.SetStateMachine(stateMachine);
        }
    }
    [StructLayout(LayoutKind.Auto)]
    [CompilerGenerated]
    private struct <GetObject2>d__2 : IAsyncStateMachine
    {
        public int <>1__state;
        public AsyncTaskMethodBuilder<object> <>t__builder;
        private void MoveNext()
        {
            object result;
            try
            {
                result = new object();
            }
            catch (Exception exception)
            {
                <>1__state = -2;
                <>t__builder.SetException(exception);
                return;
            }
            <>1__state = -2;
            <>t__builder.SetResult(result);
        }
        void IAsyncStateMachine.MoveNext()
        {
            //ILSpy generated this explicit interface implementation from .override directive in MoveNext
            this.MoveNext();
        }
        [DebuggerHidden]
        private void SetStateMachine(IAsyncStateMachine stateMachine)
        {
            <>t__builder.SetStateMachine(stateMachine);
        }
        void IAsyncStateMachine.SetStateMachine(IAsyncStateMachine stateMachine)
        {
            //ILSpy generated this explicit interface implementation from .override directive in SetStateMachine
            this.SetStateMachine(stateMachine);
        }
    }
    [AsyncStateMachine(typeof(<M>d__0))]
    public void M()
    {
        <M>d__0 stateMachine = default(<M>d__0);
        stateMachine.<>t__builder = AsyncVoidMethodBuilder.Create();
        stateMachine.<>4__this = this;
        stateMachine.<>1__state = -1;
        stateMachine.<>t__builder.Start(ref stateMachine);
    }
    [AsyncStateMachine(typeof(<GetObject1>d__1))]
    public Task<object> GetObject1()
    {
        <GetObject1>d__1 stateMachine = default(<GetObject1>d__1);
        stateMachine.<>t__builder = AsyncTaskMethodBuilder<object>.Create();
        stateMachine.<>1__state = -1;
        stateMachine.<>t__builder.Start(ref stateMachine);
        return stateMachine.<>t__builder.Task;
    }
    [AsyncStateMachine(typeof(<GetObject2>d__2))]
    public Task<object> GetObject2()
    {
        <GetObject2>d__2 stateMachine = default(<GetObject2>d__2);
        stateMachine.<>t__builder = AsyncTaskMethodBuilder<object>.Create();
        stateMachine.<>1__state = -1;
        stateMachine.<>t__builder.Start(ref stateMachine);
        return stateMachine.<>t__builder.Task;
    }
    public void MyFunction(object object1, object object2)
    {
    }
}
When diffing the results with Codebeautify.org/file-diff, I get about 8 lines of results that are very small changes. We get about 6 lines of changes between Variant 1 and Variant 2 that are basically variable name and label identifiers, and Variant 1 produces one more line of code than Variant 2 which nulls an object references within an exception block. Variant 1 has one object cast more than Variant 2 in the compiled result, but has one less null assignment.
Maybe over billions of iterations, you might see a small difference, but honestly, I think you're fine with either method.
Code readability, depending on context, is probably the biggest gain from this exercise.