He thinks he is lord Bolton, but everyone in the North knows his last name is Snow. He will be seen as illegitimate.
Roose bestowed the name "Bolton" on Ramsay as a reward. So he is legitimately a Bolton now.
If his liege lords back him as lord Bolton, then that is what he is. Roose named him as heir. Once Sansa or Bran reappear as the Stark heir that will change. I predict a grisly (and satisfying) death for Ramsey at seasons end. I love this show but the books are so much better.