From a quick search I’ve just done, the major difference is that /bin/false can’t return any text, the only thing it can do as specified via POSIX standards is return false.
So if you set someone’s shell to /bin/nologin there can be some text that says “You’re not allowed shell access”, similar to what happens if you try to SSH into say GitHub.
Of course, for a service account that won’t be operated by a person, that doesn’t matter - so whichever one you use is just whichever the operator thought of first, most likely.