Note: typing.Dict has been deprecated as of Python 3.9, because the dict type itself can be used as a generic type directly (together with other standard containers). You can do the same in Python 3.7 or 3.8 if you use a from __future__ import annotations directive. My answer was originally written for much older Python 3 releases.

There is no real difference between using a plain typing.Dict and dict, no.

However, typing.Dict is a Generic type that lets you specify the type of the keys and values too, making it more flexible:

def change_bandwidths(new_bandwidths: typing.Dict[str, str],
                      user_id: int,
                      user_name: str) -> bool:

As such, it could well be that at some point in your project lifetime you want to define the dictionary argument a little more precisely, at which point expanding typing.Dict to typing.Dict[key_type, value_type] is a 'smaller' change than replacing dict.

You can make this even more generic by using Mapping or MutableMapping types here; since your function doesn't need to alter the mapping, I'd stick with Mapping. A dict is one mapping, but you could create other objects that also satisfy the mapping interface, and your function might well still work with those:

from collections.abc import Mapping

def change_bandwidths(new_bandwidths: Mapping[str, str],
                      user_id: int,
                      user_name: str) -> bool:

Now you are clearly telling other users of this function that your code won't actually alter the new_bandwidths mapping passed in.

Your actual implementation is merely expecting an object that is printable. That may be a test implementation, but as it stands your code would continue to work if you used new_bandwidths: object, because any object in Python is printable.