I think the first game relied on weapon overheating to limit how fast you could fire.

The second game used ammo clips and that made combat far easier.

That’s right. I’ve seen arguments over which scheme is better, but the second game’s scheme is definitely more orthodox.

I also keep hearing ME1’s combat described as “clunky” and ME2’s as “smoother”. Without much experience with this type of game, I think I just don’t have the skills to feel the difference.

You're probably playing on a PC with mouse and keyboard. The Xbox 360 Mass Effect controls were not great, but that was the original release.