Many of the answers suggest how to choose only 1 option, but why not be able to use all 3? In this answer I explain how you can use asyncio to manage combining all 3 forms of concurrency instead as well as easily swap between them later if need be.
The short answer
Many developers that are first-timers to concurrency in Python will end up using processing.Process and threading.Thread. However, these are the low-level APIs which have been merged together by the high-level API provided by the concurrent.futures module. Furthermore, spawning processes and threads has overhead, such as requiring more memory, a problem which plagued one of the examples I showed below. To an extent, concurrent.futures manages this for you so that you cannot as easily do something like spawn a thousand processes and crash your computer by only spawning a few processes and then just re-using those processes each time one finishes.
These high-level APIs are provided through concurrent.futures.Executor, which are then implemented by concurrent.futures.ProcessPoolExecutor and concurrent.futures.ThreadPoolExecutor. In most cases, you should use these over the multiprocessing.Process and threading.Thread, because it's easier to change from one to the other in the future when you use concurrent.futures and you don't have to learn the detailed differences of each.
Since these share a unified interfaces, you'll also find that code using multiprocessing or threading will often use concurrent.futures. asyncio is no exception to this, and provides a way to use it via the following code:
import asyncio
from concurrent.futures import Executor
from functools import partial
from typing import Any, Callable, Optional, TypeVar
T = TypeVar("T")
async def run_in_executor(
    executor: Optional[Executor],
    func: Callable[..., T],
    /,
    *args: Any,
    **kwargs: Any,
) -> T:
    """
    Run `func(*args, **kwargs)` asynchronously, using an executor.
    If the executor is None, use the default ThreadPoolExecutor.
    """
    return await asyncio.get_running_loop().run_in_executor(
        executor,
        partial(func, *args, **kwargs),
    )
# Example usage for running `print` in a thread.
async def main():
    await run_in_executor(None, print, "O" * 100_000)
asyncio.run(main())
In fact it turns out that using threading with asyncio was so common that in Python 3.9 they added asyncio.to_thread(func, *args, **kwargs) to shorten it for the default ThreadPoolExecutor.
The long answer
Are there any disadvantages to this approach?
Yes. With asyncio, the biggest disadvantage is that asynchronous functions aren't the same as synchronous functions. This can trip up new users of asyncio a lot and cause a lot of rework to be done if you didn't start programming with asyncio in mind from the beginning.
Another disadvantage is that users of your code will also become forced to use asyncio. All of this necessary rework will often leave first-time asyncio users with a really sour taste in their mouth.
Are there any non-performance advantages to this?
Yes. Similar to how using concurrent.futures is advantageous over threading.Thread and multiprocessing.Process for its unified interface, this approach can be considered a further abstraction from an Executor to an asynchronous function. You can start off using asyncio, and if later you find a part of it you need threading or multiprocessing, you can use asyncio.to_thread or run_in_executor. Likewise, you may later discover that an asynchronous version of what you're trying to run with threading already exists, so you can easily step back from using threading and switch to asyncio instead.
Are there any performance advantages to this?
Yes... and no. Ultimately it depends on the task. In some cases, it may not help (though it likely does not hurt), while in other cases it may help a lot. The rest of this answer provides some explanations as to why using asyncio to run an Executor may be advantageous.
- Combining multiple executors and other asynchronous code
asyncio essentially provides significantly more control over concurrency at the cost of you need to take control of the concurrency more. If you want to simultaneously run some code using a ThreadPoolExecutor along side some other code using a ProcessPoolExecutor, it is not so easy managing this using synchronous code, but it is very easy with asyncio.
import asyncio
from concurrent.futures import ProcessPoolExecutor, ThreadPoolExecutor
async def with_processing():
    with ProcessPoolExecutor() as executor:
        tasks = [...]
        for task in asyncio.as_completed(tasks):
            result = await task
            ...
async def with_threading():
    with ThreadPoolExecutor() as executor:
        tasks = [...]
        for task in asyncio.as_completed(tasks):
            result = await task
            ...
async def main():
    await asyncio.gather(with_processing(), with_threading())
asyncio.run(main())
How does this work? Essentially asyncio asks the executors to run their functions. Then, while an executor is running, asyncio will go run other code. For example, the ProcessPoolExecutor starts a bunch of processes, and then while waiting for those processes to finish, the ThreadPoolExecutor starts a bunch of threads. asyncio will then check in on these executors and collect their results when they are done. Furthermore, if you have other code using asyncio, you can run them while waiting for the processes and threads to finish.
- Narrowing in on what sections of code needs executors
It is not common that you will have many executors in your code, but what is a common problem that I have seen when people use threads/processes is that they will shove the entirety of their code into a thread/process, expecting it to work. For example, I once saw the following code (approximately):
from concurrent.futures import ThreadPoolExecutor
import requests
def get_data(url):
    return requests.get(url).json()["data"]
urls = [...]
with ThreadPoolExecutor() as executor:
    for data in executor.map(get_data, urls):
        print(data)
The funny thing about this piece of code is that it was slower with concurrency than without. Why? Because the resulting json was large, and having many threads consume a huge amount of memory was disastrous. Luckily the solution was simple:
from concurrent.futures import ThreadPoolExecutor
import requests
urls = [...]
with ThreadPoolExecutor() as executor:
    for response in executor.map(requests.get, urls):
        print(response.json()["data"])
Now only one json is unloaded into memory at a time, and everything is fine.
The lesson here?
You shouldn't try to just slap all of your code into threads/processes, you should instead focus in on what part of the code actually needs concurrency.
But what if get_data was not a function as simple as this case? What if we had to apply the executor somewhere deep in the middle of the function? This is where asyncio comes in:
import asyncio
import requests
async def get_data(url):
    # A lot of code.
    ...
    # The specific part that needs threading.
    response = await asyncio.to_thread(requests.get, url, some_other_params)
    # A lot of code.
    ...
    return data
urls = [...]
async def main():
    tasks = [get_data(url) for url in urls]
    for task in asyncio.as_completed(tasks):
        data = await task
        print(data)
asyncio.run(main())
Attempting the same with concurrent.futures is by no means pretty. You could use things such as callbacks, queues, etc., but it would be significantly harder to manage than basic asyncio code.