They're total opposites. Dogs are friends and companions, who generally aren't all that interesting when you're not interacting with them. Cats are more like housemates, who do their own interesting shit all on their own.
Like, with my dog, I can actually get on the floor and play with him. I can wrestle with him and toss him around, and we're actually interacting. My cat on the other hand just lies around all day, but this morning she ran out the door and went on her own little kitty adventure for a few hours and then came back.
I think having both is a nice mix.