To me hard shell means it's waterproof and doesn't breathe;
"Soft shell" is a recent coinage; "hard shell" is the opposite of "soft shell". Unfortunately the softness of the shell has very little to do with the breathability or waterproofness of the garment; that's determined by the presence (or absence) of a waterproof layer below the outer shell. In my mind, "hardshell" means fully waterproof or else hard and not breathable, "softshell" is soft, is not waterproof and is very breathable.
(So my ancient department-store woven-nylon snow pants are hardshells: they're not soft or breathable at all. They're merely water-resistant, especially now that the water-repellent coating has mostly worn off, but they're defintely not softshells. Old, dirty gore-tex that has lost its waterproofness but has a soft outer layer would fall into the same category: not very breathable, not waterproof.)
Waterproof gear originally was oiled canvas or similar: completely non-breathable, with a smooth outer finish (extremely dense weave or a continous membrane, like rubber). Modern "hardshells" often have a water-repellent woven outer layer, and a "breathable" waterproof membrane (Gore-Tex or eVent) underneath. I put breathable in quotes because they don't breathe all that well. You can find graphs of transpiration rates and you'll see that Gore-tex barely breathes until the inside is thoroughly swampy. EVent starts working measurably lower on the temp/humidity curve, but I think you'll still feel more like you're wearing a raincoat than like you're wearing, say, a light fleece.
Softshells (by my definition) have a looser outer weave, and no inner membrane, so they actually do breathe. They're also not really waterproof; the outer weave is water-repellent but won't last long in a really hard rain. But they're great in winter, they shed snow like anything and they're good enough to keep your knees and elbows dry while ice-climbing even on somewhat damp ice. Wind resistance is generally not as good as with hardshells, but this varies.
If you normally climb with a softshell over a base layer, climbing with a fleece plus a hardshell (eVent or otherwise) will probably make you sweat.
Personally I don't have a softshell jacket (though I love my new softshell pants); I often climb in just a fleece. In addition to my belay jacket I often carry a hardshell in case of high wind or spring rain. It usually stays in my pack, but it's a small one and doesn't weight much. If I had a softshell jacket, I'd probalby leave the hardshell at home except if I feared rain.