"Disclaimer": I played through Mass Effect right before Mass Effect 2, so this is mostly a comparison between the two. I played both on the PC (I also played the first one on Xbox 360, 2 years ago when it came out). No spoilers. Also, whatever I say here, I loved the game and would recommend it to anyone who played Mass Effect and liked it.
Missions. My god, the quality of missions is orders of magnitude above Mass Effect. They completely took out the Mako missions, but you don't miss them at all. The mission environments are for the most part all different, no more repeats of the same facilities over and over again. Character involvement is much higher across the board, especially interesting are the "loyalty" missions. Don't pass these up.
Graphics. The graphics got an overhaul for the sequel. Environments are prettier, abilities are better looking, and there's generally more polygons all over the place. I'm running the games on an Nvidia 9800 GTX+, an intel dual 3.0 Ghz processor and 4 gigs of ram @ 1650x1080.
Inventory and upgrades. They changed these around, for better or worse. I generally liked the changes. Instead of carrying around a 150 item inventory, you carry around 2-3 of each type of weapon (with six different heavy weapons options, yeah, now YOU get to carry a rocket launcher). The attachement items from Mass Effect are replaced by ammo talents, and damage upgrades are researched on the ship using resources you mine (more on that in the next section).
Elevators. They got rid of the elevators!! This was my personal pet peave in Mass Effect. Stepping into an elevator in between areas, getting locked into place, and waiting interminably without ever being able to save, stuck watching the wall move sloooowly past. There's still a few elevators in the sequel, but nowhere near as many, you can save in them, and the most important one, on your ship, you don't ride, you just cut to a loadscreen.
Resource gathering. This sucks like an industrial strength vacuum cleaner. In order to pay for upgrades to your ship, weapons, shields, health, etc, you have to pay with resources. You can find a few resources lying around in boxes on missions, but to pay for the serious upgrades, you have to go probing. Instead of hitting the scan button and finding some stuff like in Mass Effect, you have to manually scan the entire planet. You hold down the right mouse button and slowly wave your mouse across the planets surface, watching until your scanner spikes. Then you wave it around some more trying to find the highest concentration in the area. Then you launch a probe at the surface and collect a couple thousand units of a resource. You need 50,000 units for the bigger upgrades, and there are a lot of upgrades to collect, most of which are very beneficial. Oh yeah, and you have to pay for the probes. And you have only enough space for enough probes to survey half a system. (New pet peave to replace elevator fixation)
Glitches. I'd occasionally be walking along, then suddenly be stuck in place 3 meters up in the air. Once I glitched into a cliff. It only happend a few times, and mostly on side missions, so it didn't turn me off the game, but it still jerked me out of my focus pretty hard.
Clipping. I think that's the problem when textures push through each other. This occured in a couple of cutscenes, mostly on faces, where somebody blinking would have white textures showing popping through their eyelids, or their neck would go through their shirt collar. This was only really bad and noticeable on one or two occasions, but it would happen to minor details fairly frequently.
Captain Anderson. He wasn't exactly good-looking in the first game, but the facial update made him downright ugly. He made one expression that made me recoil in disgust. It wasn't pretty.
In conclusion: I loved the game, and will probably play through it again as soon as I get over being throughly annoyed with the resource gathering system. If I could cheat to get around that, I would.