Saints Row 2 is the biggest game in the series. It has more activities, more characters and more missions. Its open world is larger and more complicated. There are more character-creation and clothing options. There are more vehicles. It's just all-around more.
However, just because it's the biggest doesn't mean it's the best. Don't get me wrong, I wish Saints Row: The Third had as much to do as Saints Row 2. I think it would have been an unbelievably stellar game if it had expanded and refined the series open world playground. To have SR2's variety with SR3's narrative confidence would have been near-perfect, but if I have to choose between the two, I'm going to go with narrative confidence.
Maybe that's backwards. Maybe I should be judging the games purely by gameplay (though SR3 is no slouch in that department), but the best part of any Saints Row game is the writing. The characters and the setting bring a sense of fun that is often missing in other open-world games (which is why I'm not presently writing to you about Grand Theft Auto V, which is technically superior in every way and an amazing achievement in world-building, but so mean-spirited and up its own ass that I'd sooner play the original Saints Row, and that game barely worked).
This holds true for Saints Row 2. I've criticized the writing for being uneven, but that unevenness is really just a matter of tone. A lot of my favorite moments in the game were in cutscenes. I loved seeing Pierce get the short end of the stick time and again. I loved Shaundi's seemingly-endless supply of ex-boyfriends (seriously, where does this chick get the time to be a drug dealer). I loved Gat's cheerful psychopathy. Dane Vogel trying to fast-talk his way out of an unrighteous beat-down. Maero and the Boss being basically the same person. The endless, absurd corruption from just about everyone who gave you a side-mission. These are all great. It's just unfortunate that they are tied in with a couple of main stories that were too serious to fit in (though they would have been fine on their own right).
In the end, I tend to regard Saints Row 2 as a prologue to Saints Row: The Third. Not to downplay it or anything. It is a necessary prologue. I just like being able to see the origin stories of Pierce and Shaundi, the start of Gat's obsession, and how the Saints rose to power before becoming world-famous superstars.