In American English, stuttering is the more commonly used term, while in countries such as England, Australia, and South Africa, the preferred term is stammering.
There does seem to be a difference in how the terms are used once one steps out of the speech and language situation. While both can be used to describe any unintentional repetition of sounds in speech, stuttering can also be used to describe unintentional stopping and starting of video games or a difficult beginning of a sports game or season.
Again, these terms mean exactly the same thing, and usage is mainly determined by one’s country of origin.
Neither! You’re a person. You may be a person who stutters (PWS), but you’re not defined by your speech.