The difference between them is that “Матвей” and “Матфей” are Russian analogues of English “Matthew” while “Мэтью” or “Мэтт” are the same “Matthew” and “Matt” but written with Russian letters. Their purpose is just to transfer English sounding.

By the way, same with your name. “Николай” is Russian analogue for English “Nick” while “Ник” is not a Russian name. It is the same word “Nick” but written in Russian.