'Cute' makes me think of little brother cute as in good looking but I want to hug him and eventually tell him a bed time story or something. Though I dislike the word, 'handsome' makes me think of good looking in a shagable sense. Like Gerard Butler. I would let Gerard Butler abuse my body.
Agreed that personality is an important factor, but asuming they would both be lovely guys, I'd go for strong and handsome.