Both terms refer to a relationship that is not yet in the serious stage.
If you say you are dating or seeing someone, you are inferring that you are going out with someone with at least some regularity.
"Seeing someone" means you're a sicko-pervo stalker voyeur dude.
Let's use what we learned in a sentence, "While Da Ge was dating a girl he went to college with, little brother Booby was regularly seeing both his sisters in the shower through a secret peephole." actually depend on people...because seeing someone or dating someone just not that serious for "zhanan"...u know ?
In that case, you would probably say that you have a boyfriend (or girlfriend), or you're in a 'relationship' or have a partner, etc.