Cute means attractive but it can also apply to a cute personality in which the guy has a lot of likable traits like he is really nice or funny. But cute can be almost childish...like a guy with a baby face or a boy next door look to them. Cute guys are more approachable and seem to have a friendly aura to them.
Handsome is more of a unanimous agreement that you are very attractive. Most movie stars are considered handsome. And it's more of a more distinguished and masculine look. Some qualities that come to mind are a distinct jaw line, muscular body or just classic good looks that most women would agree is hot! However, handsome men aren't as approachable because they can seem intimidating and hence, unapproachable. Yet, handsome men are usually the guys women fight over to get.
Being cute is great but if you are certain that you are in the handsome category then you are one lucky guy. Women love that because its rare. You probably have no problem getting women. Um, not to sound like a pervert or weirdo but would you mind letting me see pic of yourself? I normally wouldn't ask but I'd like to see for myself...lol.