I'll list the good things about it first, as there are few of them.
The combat is EXCELLENT and greatly improves upon the previous games. Combat in Saints Row 1 and 2 felt disconnected and fairly shallow, but Saints Row 3 adds in better gunplay and an improved melee system, as well as upgradeable weapons. There are new "specialist" enemies with different abilities that mix things up.
There is a new upgrade system in which you can level up and purchase abilities, like improving your sprint stamina and allowing you to dual-wield weapons. While most of the "upgrades" were either available from the start of Saints Row 2 (like dual-wielding) or unlockable through activities (such as bullet resistance and increased sprint stamina) they do introduce some new ones, which give a certain sense of accomplishment and something to work towards as the game goes on.
Lastly you can now upgrade your weapons. I'm mixed on this feature, seeing how all of the weapons are incredibly weak from the beginning and only become about as powerful as their SR2 counterparts when mostly or completely upgrade, but overall I like it. Being able to set enemies on fire with an SMG is great fun and reminds me of some great times playing Borderlands.
Now for the cons...I don't want write out every little gripe I have with this game, so I'll just list them off. As mentioned before, my main complaint with SR3 is that it doesn't measure up to Saints Row 2 in most ways and really does feel like less of a game. Accordingly, most of the complaints are regarding things they removed from the previous games.
-the new city (Steelport) isn't as well designed and doesn't have nearly as much variety or detail as Stilwater had
-nearly everything can be accessed from your phone or crib, leaving no need to drive around to Rim Jobs, Hit-List locations, etc. (this makes a little sense as, like I previously mentioned, the new city isn't nearly as detailed or varied, making cruising around not as fun)
-activities no longer have their own cutscenes (remember the fun intros to things like Hitman and Insurance fraud), most are instead introduced to you through a story mission with no real explanation as to why you are doing these things
-for some reason, after upgrading a weapon in your crib, every weapon of that type that enemies drop becomes your upgraded version
-there are maybe 1/4 of the clothing options Saints Row had
-clothing can no longer be layered; you have to choose between preset shirt/jacket combos
-most of the businesses are gone, including (but not limited to): all fast food stores, all drug/alcohol stores, and all of the "regular" clothing stores (not Nobody Loves Me, Let's Pretend, or Leather & Lace)
-all of the "normal" clothing stores have been replaced with Planet Saints, a rather obnoxious Third Street Saints-themed store that is inexplicably set up all around Steelport (where the Saints are hated by virtually everyone and where they have never done business before)
-the hairstyles aren't as varied as SR2, and quite frankly most of them just suck
-you can't customize cribs anymore
-there are less weapons, which the developers try to justify with weapon upgrades
-the AI is probably dumber than ever, with enemies just standing in the open and comrades regularly getting stuck on objects or running into walls
-there are no more consumables
-the Rim Jobs menu is much more awkward than before; the camera is set at weird angles unless you're selecting paint, resulting in some options not being visible unless you buy them and drive out
-the radio stations aren't as fun (and in my opinion, most of the songs suck)
All in all Saints Row 3 is a solid game...it's fun, there's a ton to do, and there's a whole lot of replay value. I just can't help but be disappointed in Volition's removal of so much content and to me, it feels like a ripoff. There isn't nearly as much new content to justify the removal of so much.
If you're new to the Saints Row franchise it's probably worth picking up Saints Row: The Third.
If you've played Saints Row 2, you'll probably be disappointed unless you start the game with a clear understanding that this is NOT the same thing. Ignore the names of the main characters and think of it as a new game entirely and you'll probably enjoy it.