It's a difference in style of play more than it's a direct upgrade over Casspi and Greene. Casspi is very active defensively but he's not going to physically dominate anyone. Similarly, Greene is a smooth athlete and he moves his feet well but he's not particularly strong for a player his size. He would have trouble guarding PFs or physical post-up small forwards. Aminu is a guy who could guard three different positions and have a physical advantage at all of them most nights.
And for all the talk of his weaknesses on offense, he still averaged almost 16 points per game last year. You can question how much of his scoring came close to the basket and whether he can sustain that against taller defenders. He's not anywhere close to an outside threat right now, but he's already working on his jumpshot. He won't be a prototypical SF but Josh Smith is a pretty effective player offensively since he stopped shooting all those threes last year so the range isn't a must. Especially if you're a little creative about how you manage your rotations.
So, like hammystyle said, Aminu's value right now lies primarily in his defense and rebounding but I don't think he'll be a black hole on offense either. He's a step down in talent from Derrick Favors, but has a similar type of potential. And remember that despite being a sophmore, he's in the same age range as John Wall and Demarcus Cousins -- younger than Wall by a couple weeks even.
I still think Cousins has to be the pick if he's there, but between Johnson and Aminu -- they both offer different skillsets. Wesley Johnson looks like a very good compliment to Tyreke offensively and at least a capable defender. And he's a lot closer to reaching his potential than Aminu is right now so he's far far less risky. Aminu looks like a very good compliment to Tyreke defensively and at least capable on offense, with the caveat that it's going to take some time and some dedicated effort to get him there. Personally I don't think it's a no brainer choice between the two, but it seems like Petrie tends to go with "sure thing" over "raw potential" most of the time.