I feel it less than I ever felt a tampon. I agree with the woman who said that it might be nice for it to be *less* comfortable, since sometimes you can forget it's in there!
FWIW, since people asked about cloth backup, I also find slim cloth pads to be more comfortable than disposable pads. All-around, reusable products are just nicer. They smell less, they're more comfortable, and there's no waste. Plus, it is SO cool not to ever run out of stupplies, or have to remember to pick up tampons at the grocery store. It just occurred to me that I haven't bought tampons or pads in over 3 years.
Mess-wise, I don't think it's bad. On light days there's virtually no mess. On heavy days I sometimes do get a little blood on my fingers but that can be wiped off with TP and then washed off in the sink. I would sometimes have the same thing happen with tampons, so I don't think of that as a down side of a cup. On the rare occasion that I would have to dump it in a public restroom, I've never felt like there was a gross mess to deal with. I do think a cup is more discreet for public use than a pad or tampon - no "hey, everybody, I'm having my period!!!" rustling of paper or plastic in your stall.

It's also nice to feel like you're doing your part to keep the stink down in the women's restroom, since you're not leaving smelly soiled menstrual products in that little garbage bin next to the toilet.