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.

This is a recursive function. When you reach the terminal case (string == ""), you return 1 or 2. That gets returned to the calling function -- the previous call of nfsmsim. But that call of of nfsmsim doesn't return anything! You need to get the value from the terminal call of nfsmsim and pass it on by returning it again.

In other words, you need a return statement in each of these two branches of your if statement:

elif (current, string[0]) in edges.keys():
    global loc
    string = string[1:]
    nfsmsim(string, loc[0], edges, accepting)
elif len(loc)>1:
    global loc
    nfsmsim(string, loc[1], edges, accepting)

Might not be exactly on topic but I wish more people would code their functions to NOT return None and instead return expected type.

BeautifulSoup is the biggest perpetrator of this. Whenever you parse the tree/soup you'll either get a list of results or a single result, now if no results are found instead of getting an empty list you get None returned, which in turn requires you to make checks on every single lookup which really messes up your code up and makes everything look ridiculous.

switched to lxml/xpath and never looked back.

Because your summ function only returns a value if nums is an empty list.