Equally as good as the second
I still haven't played the first game in the Saint's Row series, and I'm not quite sure I should bother, but the second and third entries in the series are certainly masterpieces. This is a game that is completely committed to providing the player with batshit insane FUN and it does it well. Almost any sensibilities of seriousness have disappeared and this is easily the silliest sandbox game ever made. I liked Yahtzee's comment that this is a game on full-on "Evil Dead" syndrome (first one was serious, second a little funny, third completely ridiculous).
That said, I don't necessarily think its an improvement. Just "equally good". This is a well-polished game, but it feels a little unfinished in parts (that's not to say an immense amount of effort hasn't gone into this game, they've had to record six different voice actors after all!) whereas Saints Row 2 felt a lot more cohesive. SO I strongly recommend playing both games, and together they will provide you with hours of entertainment. Oh, and also, if you're going to get any DLC, the "Trouble with Clones" is the best one.