This is my favorite PSP game, and, in my opinion, a fantastic, fully legitimate extension of the Final Fantasy VII world.
The story in the game was what I was really looking forward to, and I feel that it delivered immensely.The impetus for Zack's journey is that he wants to become a hero. That's it! That's all. Sounds pretty boring, right? Well, the game does a much better job with the story it isn't trying to tell than the one that it is. SOLDIER's internal strife, seeing the upper plate of Midgar, seeing what SOLDIERS do both on and off the job and giving them a face and a purpose, showing the contrast between slum life and corporate life, showing us how Shinra screws up your family life - I love how Genesis had normal parents and it was good guy Angeal's dad who was messed up - giving Hojo a scientific rival where he was the only scientist at all in the original in a world where that's pretty important (besides Gast, who was dead), showing us a bit of the war with Wutai that was only briefly mentioned, showing us more of Cloud and what he was like as a Shinra grunt, showing us the relationship between Aerith and Zack that was briefly mentioned in the original game, etc.
Crisis Core did a really good job at showing us the human Sephiroth, too. I love the line when he's talking to Zack and Zack says something like "For real!?" and then Seph laughs and says "For real." That's something we'd never see him do after his fall. It was the same with talking about his past and having friends. Also, there was Tseng and Aerith's relationship, which felt tacked on in the original FFVII but more fleshed out as something innocently one-sided in Crisis Core. Tseng himself also got a lot of focus, which was nice, as he was absent for a good portion of VII.
As an aside, Before Crisis showed us more of the Shinra hero side of Sephiroth. I wish that this game was localized somehow, as it really was a huge part of the compilation. While CC shows us what's going on in SOLDIER, it's pretty compartmentalized in that regard. An equally important story, Shinra vs. AVALANCHE, happens concurrently in BC, and also fills in the gaps from when Zack was knocked out in a tube in Nibelheim. I would even call it the "main" story of the prequels, whereas CC is, like I said, more of a side story.
By far the best parts of the game were seeing Zack and Cloud interact with one another and what Sephiroth was like pre-insanity. Zack, and SOLDIER in general, really, were such huge parts of the FF7 lore, and it was so satisfying to see them finally fleshed out in a game. Zack (the very brief glimpses of him we see) was portrayed in Final Fantasy VII as a headstrong, friendly, outgoing guy, and his Crisis Core self was the perfect reflection of that. Cloud was not always the cocky and arrogant man that he begins as in VII, originally being a meek and mild boy. Crisis Core shows us more of this side of him, his original personality, if you will, and you get to watch him gradually become attached to Zack as a sort of older brother or mentor figure.
The "become a hero" story was basically fluff to show us all of that. At least, that's how I see Crisis Core. It's a side story. It's more of a companion than its own thing. A reason for us to return to and see more of Final Fantasy VII's world and characters, which the game did and did well.
I totally understand people being upset with the over the top ending and key word: hero nature of the story, but the first is a matter of taste (I very much enjoyed the ending to this game, and the playable segment was amaaaaazing) and the second I already commented on, where the background story is more interesting than the story about becoming a hero, which is really only a vehicle for Zack to experience the things that he does.
Zack's death in the original VII flashback was short, quick, and violent, a testament to the brutality of Shinra and the world that the VII characters inhabit. Zack's death in Crisis Core is full of flair, and is really quite the opposite of its original rendition in terms of tone, even though it keeps the final segment where the three Shinra grunts overtake him. It was a bold decision for Square to change the ending the way they did, but I personally love both versions of it.
Oh, and the soundtrack is amazing. The World's Enemy is the creepiest rendition of One-winged Angel there is in a game. Dreams and Pride accurately represents the starry eyed, ambitious young version of Zack that we start the game as. Moonlit Wandering has that melancholy country vibe to it, perfect for Zack's escape from the Shinra mansion. Howl of the Gathered plays during battles in the final dungeon, reflecting Zack's desperation to return to Midgar. Even better is that renditions of some of these tracks were originally heard in the Last Order OVA, which, by the way, is mostly canon to this game and shows events that the game itself skips over, the two biggest ones being Zack on the highway and showing more of the truck scene at the end.
It will be interesting to see how much of this game makes it into the VII remake. Perhaps the most highly divisive element of Crisis Core was the character of Genesis, who shared the voice and likeness of Japanese pop star Gackt. The reason people took issue with Genesis wasn't because of his appearance or voice, however - the man had a tendency to constantly recite poetry in place of actual dialogue, played an extremely integral role in a story where he didn't exist in prior VII lore, and even shoehorned his way into cutscenes depicting events from the original game, such as the Nibelheim flashback. My opinions on Genesis are divided. He is more appealing conceptually than he is as he is portrayed in game, as a friend of Sephiroth and fellow SOLDIER First Class. His nature as a genetic experiment might be a bit copy-paste from Sephiroth's own story, but I feel like it has a realistically believable in-universe explanation, as Shinra is wont to do these sorts of things since way back in the timeline - see: Vincent Valentine's flashback, over fifty years before the main story. He certainly had no place in the Nibelheim scene, but other than that I think he functions well enough as the primary villain of the game. Interestingly enough, Gackt's contract is rumored to be the most likely reason that the game hasn't seen a digital re-release on the PlayStation Network, so Genesis's inclusion in the VII remake remains largely up in the air. I think we can say for certain we'll be seeing more of Zack, at least!
One last comment about the soundtrack and the ending: perhaps a divisive opinion, but The Price of Freedom is the best damn final battle music in any game, ever.
This topic is a stub of an article I wrote back when the VII Remake was announced. I was still overly wordy and unprofessional back then, but here's the whole thing, if you want to read it.