On the actual behavior, there is no difference. They all return None and that's it. However, there is a time and place for all of these. The following instructions are basically how the different methods should be used (or at least how I was taught they should be used), but they are not absolute rules so you can mix them up if you feel necessary to.

Using return None

This tells that the function is indeed meant to return a value for later use, and in this case it returns None. This value None can then be used elsewhere. return None is never used if there are no other possible return values from the function.

In the following example, we return person's mother if the person given is a human. If it's not a human, we return None since the person doesn't have a mother (let's suppose it's not an animal or something).

def get_mother(person):
    if is_human(person):
        return person.mother
    else:
        return None

Using return

This is used for the same reason as break in loops. The return value doesn't matter and you only want to exit the whole function. It's extremely useful in some places, even though you don't need it that often.

We've got 15 prisoners and we know one of them has a knife. We loop through each prisoner one by one to check if they have a knife. If we hit the person with a knife, we can just exit the function because we know there's only one knife and no reason the check rest of the prisoners. If we don't find the prisoner with a knife, we raise an alert. This could be done in many different ways and using return is probably not even the best way, but it's just an example to show how to use return for exiting a function.

def find_prisoner_with_knife(prisoners):
    for prisoner in prisoners:
        if "knife" in prisoner.items:
            prisoner.move_to_inquisition()
            return # no need to check rest of the prisoners nor raise an alert
    raise_alert()

Note: You should never do var = find_prisoner_with_knife(), since the return value is not meant to be caught.

Using no return at all

This will also return None, but that value is not meant to be used or caught. It simply means that the function ended successfully. It's basically the same as return in void functions in languages such as C++ or Java.

In the following example, we set person's mother's name and then the function exits after completing successfully.

def set_mother(person, mother):
    if is_human(person):
        person.mother = mother

Note: You should never do var = set_mother(my_person, my_mother), since the return value is not meant to be caught.

StackOverflow has a good discussion about this exact topic in this Q&A. In the top rated question, kronoz notes:

Returning null is usually the best idea if you intend to indicate that no data is available.

An empty object implies data has been returned, whereas returning null clearly indicates that nothing has been returned.

Additionally, returning a null will result in a null exception if you attempt to access members in the object, which can be useful for highlighting buggy code - attempting to access a member of nothing makes no sense. Accessing members of an empty object will not fail meaning bugs can go undiscovered.

Personally, I like to return empty strings for functions that return strings to minimize the amount of error handling that needs to be put in place. However, you'll need to make sure that the group that your working with will follow the same convention - otherwise the benefits of this decision won't be achieved.

However, as the poster in the SO answer noted, nulls should probably be returned if an object is expected so that there is no doubt about whether data is being returned.

In the end, there's no single best way of doing things. Building a team consensus will ultimately drive your team's best practices.