Trad Climbing harness
To add to Ben Crowell's answer, some additional differences between mountaineering and rock-climbing harnesses include comfort while hiking, and weight.
Compare the two harnesses below; the first is for alpinism/mountaineering, it's simple, light, and very minimal in size and bulk. A harness like this would be extremely comfortable to hike in, and wouldn't be that uncomfortable to wear with a backpack, you can see how quick and easy it would be to put on and take off as well. There are some harnesses on the market that are even more minimal than this one, they aren't designed to be comfortable while being sat in, they're just supposed to be there to protect you incase you or one of your team slips on a traverse or falls into a cravass.
The next harness is at the other side of the spectrum, this is a big wall harness made for being sat in for extended periods of time and carrying lots of gear and tools. This harness would be very comfortable to sit in and hang all of your pro off of, but it's pretty bulky, so if you were to try and hike in it you'd notice it a lot, and would probably chafe your legs and it'd be really uncomfortable under the waist belt of your backpack.
If you're planning on getting pretty serious into both activities I'd recommend getting a good trad harness (the big wall harness I show above is super overkill, better for aid climbing than most trad), and then buy a second harness for mountaineering (or just carry some webbing like Ben suggests). One big advantage to buying a light and simple alpine harness, is they're also a lot cheaper, so you can actually get one for cheaper than one of your cams.