Ramsay Bolton is a psychopath, i.e. a person who isn't honest nor ashamed of anything, and has no conscience. Psychopaths also tend to wage psychological warfare. That's what it is, he wants to hurt Jon and belittle him, trying to make him feel bad and inferior.

Also, he is a bastard himself and that is the thorn in his side which stings the most. In a feudal society, bastards can't inherit anything, and even if they are legitimized, they are somewhat looked down upon and are still considered to be "not quite as good as" legitimate children. In the books, this freaks him out even more, partially because Roose Bolton had a legitimate son (Domeric Bolton) who predeceased him.

As someone being belittled for his bastard status, I think he feels like he can finally belittle someone else (jon), and that feels good to him. Overall, I'd say it's a mixture of inferiority complex and psychological warfare.