I use them all interchangeably. My boyfriend hates it when I call him cute, but I really do it because he's done something sweet. I'm not trying to put him down, I'm just trying to compliment him on how he is being really sweet :)
But I use cute, hot and handsome interchangeably. Cute can, for me, have two meanings. The sweet one and then because I find him attractive.
When I say a guy's cute, it's usually because something he does is "cute" (i.e. sweet, awkward, romantic, etc.) or if I really like a guy's personality and then notice that he's attractive, I'd say he was cute (but usually only if his personality dominates my attraction for him).
Handsome is an extremely sophisticated guy. Really put together, fairly serious, almost gentle-looking...the kind of guy that is attractive but not in that animalistic, sexual way. It can be, but not usually. For me, handsome is the guy I'd glance at across the room and appreciate but not necessarily want to touch.
Hot is, "oh my god, I want to get in his pants now" lol. I'll call a guy hot before I've ever said a word to him. It has nothing to do with personality, but all to do with rugged good-looks.