Yes, while both can sense your mood, seemingly knowing when you're sick or feeling down, a dog will come up and try to comfort you. A cat, sensing the same, will just say, "So--I don't give a crap."
Somewhat true, but not always. I have had two cats who showed considerable consternation and concern when I was ill.
I think they show it differently than dogs do, when they do care.
Any time I wind up on the floor, for any reason, the cats come running. I’ve always found the myth of the aloof cat interesting. I’ve owned many a cat in my time, and they’ve all been downright clingy. I think it’s more about the owner than the animal. If you want your pets to be more ornamental than companions then you get aloof animals, if you’re looking for companions you get lap dwellers.