As explained by everyone, functionally there's no difference between list.Count != 0 and list.Count > 0 as list.Count can not be < 0.
I did a quick test, and it shows both != 0 and > 0 are almost equally fast (and that's super fast). Linq's list.Any() is NOT as fast though!
Here's the test code (comparing 100000 times to magnify the difference)
static List<object> GetBigObjectList()
{
    var list = new List<object>();
    for (int i = 0; i < 10000; i++)
    {
        list.Add(new object());
    }
    return list;
}
static void Main(string[] args)
{
    var myList = GetBigObjectList();
    var s1 = new Stopwatch();
    var s2 = new Stopwatch();
    var s3 = new Stopwatch();
    s1.Start();
    for (int i = 0; i < 100000; i++)
    {
        var isNotEqual = myList.Count != 0;
    }
    s1.Stop();
    s2.Start();
    for (int i = 0; i < 100000; i++)
    {
        var isGreaterThan = myList.Count > 0;
    }
    s2.Stop();
    s3.Start();
    for (int i = 0; i < 100000; i++)
    {
        var isAny = myList.Any();
    }
    s3.Stop();
    Console.WriteLine("Time taken by !=    : " + s1.ElapsedMilliseconds);
    Console.WriteLine("Time taken by >     : " + s2.ElapsedMilliseconds);
    Console.WriteLine("Time taken by Any() : " + s3.ElapsedMilliseconds);            
    Console.ReadLine();
}
And it shows:
Time taken by !=    : 0
  Time taken by >     : 0
  Time taken by Any() : 10