Saints Row 2 felt like a spiritual successor to the PS2-era GTA games. It took a bunch of the bad/grindy things from San Andreas and made them fun, and upgraded fun stuff that never made it to GTAIV. Unfortunately, the PC port of SR2 is terrible. It lags worse than SR3, despite the graphics being not as good. That said, I haven't played with the Gentlemen of the Row mod, which allegedly fixes the lag issue, but I had a blast with it on PS3.

You want new clothes? Go to the mall, they have a bunch of clothing stores.
You want to change your character's appearance? Don't bother with the barber shop, gym, and food, just go to the plastic surgeon. You can go from a skinny young black man to a fat old asian woman in less than 2 minutes and for only $500 or less.
Gang territory improved over San Andreas, so you don't actively lose territory, but you still have to defend it.
Car modifications are better. Also, if you abandon your car (or, more likely, a mission takes it away and makes you drive something else), you don't lose it! It's still in your garage! The garage limit is also bigger, and you don't have to go to the garage you last parked it at. All garages are connected.
Side-quests/minigames give you perks and abilities. Maybe this is less-known, but in the older games, you got perks for ambulance, taxi, police/army (vigilante), and firetruck optional missions. Saints Row 2, they're more fun, have more character, and give better and more unique perks.
They brought back rampages!

For me, Saints Row 2 is where the series shined. The juxtaposition of these serious characters and a serious story in this wacky world made it great. I haven't played the first, but I've heard that it's all-over an improvement. 3+ made the story less serious, became too self-aware, but the gameplay was solid and fun. I enjoyed it, but it lost some of that magic that made 2 so good.