I mean, theres a difference between lying about your name and having a false identity. A fake identity, people are greeting you down at the pub, merchants will take your credit, maybe you can get yourself an invite to a fancy/private party. Stuff like that.

Also worth pointing out, a false identity doesn't mean you are disguised. You might still look like you, which can be advantageous because people can't see through your real face, or expose your beard as fake if its your real beard. Wear a disguise, commit your crime, then take off your fake beard and turn back into Brice Lane, mild mannered playboy and new money in town, who everyone in town will vouch for being an upstanding chap.