You'll do everything an open world shooter should do, maybe a little more, but not by much.
Well first off I should express that I played both the original release and the Don's Edition, so I'll touch on a few compare and contrast subjects.
First of all is the graphics. There's a definite texture difference between regular world and your cutscenes. Most cutscenes look authentic, sound authentic, and feel authentic. All in all they're pretty good, even by today's standards. However, even in the original release, the open world always looked bland. The Don's Edition is no better, despite it's boasts about having graphical updates. It's just the monotonous brick and yellow limestone buildings that get to me, after a while it all ends up looking the same.
Luckily, despite the blandness of the world, fighting the people in the world is a delight to be had. Whether kicking an innocents butt, killing some mobsters, or sending the police insurance through the roof, (haha) It's the tense action packed fun that will keep you coming back to it.
Another great part about this game is your customizable character. In fact I could almost forgo everything else in this game just to do this over and over and over again. This was one of the first games to really do customization good. I've had fat characters, scarred characters, wrinkled old characters, the works from that creator, and on top of that you get to customize your characters suits from an array of styles and colors. In short, don't take this lightly, you'll rarely see options this fleshed out.
Interiors however are another problem. They're better than the exteriors of course, but after you've seen one, you've seen seen them all. It's not as frustrating because usually you're on a mission or taking over the joint when you enter.
Story is a good point to this as well. It's true blue Godfather Part 1 stuff, you'll recognize certain scenes easily, and be eager to fight for the family. It truly feels like you were a major part of the movies, without making any cheap changes to fit this storyline. Yep, you were always there, just a minute before or after the camera shot the scene.
All in all, if you like story, the Godfather, and customization, definitely check this out. If you like open world freedoms, you'll want to stick with GTA instead.