In addition to what has been said by others, soft shells can be tighter fitting, while allowing a greater range of motion, and offer more friction on rock and ice. However, unless you actually have a "problem" with your hard shell (I.e. too much sweat retention), I wouldn't feel compelled to buy a soft shell.
For features, I recommend researching the "gamma" series of soft shell pants from Arcteryx. The various models have pretty much all the bases covered (side pockets, low bulk belt, lace hooks, reinforced high wear areas, etc.). I have the gamma LT pants, and I use them for EVERYTHING (summer alpine, winter alpine, ice, rock, hiking, trail running, yoga). My hard shells haven't seen the light of day since I bought them.
Puffy pants are definitely a worthwhile investment. I know that Montbell and arcteryx make suitable puffy pants. The key feature is full side zips, so you can throw them on, while wearing crampons and a harness. I also like having a fly that allows my belay loop to stick out, while keeping my hips insulated.