First off, I am a fan of FF7. Even have it on emulator so I can still play it if I feel like it. Just so ya know where I stand with it.
I would agree that a large part of its popularity is the nostalgia. It may not be so great by modern standards, by any means; gameplay, graphics, exploration, but when it came out it was one of the first 3D RPG's around, I think it caught much attention for that.
Also about the graphics, and not that it changes anything about them, Tetsuya Nomura had always wanted the characters to look as they do in the movie Advent Children (obviously not like sausage people). I think they should re-release FF7 with modern day graphics and have the characters look as they were intended to.. but I digress.
The story was the big point of the game, as it is with all JRPG's, of course; you don't play them for the oh so "exhilarating" battle systems. The story was entirely driven by it's characters, and for me personally (nostalgia effects I mentioned at the beginning) when I first played it, most of those characters were something new to me and different from each other. And also that the story was not purely and simply one plot, every character had his or her own story and motivation which made them feel like a lot more than Mindless Obedient Ally Drone #3.
As an overly sheltered kid myself being brought up in the early ages of our "PC world" (not personal computer, the other PC) FF7 was one of my first exposures to any kind of heartbreak. Speaking for myself, I became very attached to some of the characters as the story went on, and by the point that *SPOILER ALERT* Aeris dies, DIES, not "oh bother, I seem to be badly wounded." Not "alright alright let's just revive the ditz and move on." No, but dead, actually and properly DEAD, I felt the tragedy. It was heartbreaking, and that, I feel, is where a lot of the nostalgia comes from. *SPOILER ALERT OVER*
Last point, the bad guy.
The glory of a hero is defined by their villain. Sephiroth isn't a typical money and authority bad guy, nor is he a run of the mill devastation type, his motives run a lot deeper than just mindlessly wanting to blow up the world. A bad guy isn't so evil if he understands morality well enough to KNOW he's a bad guy, but when a bad guy does bad things because he THINKS it's a morally righteous thing, that's when that bad guy's gone a tad crazy. And Sephiroth is totally psychotic. He's deeply emotionally perturbed over events surrounding his existence and his mother.
Aside from the emotional aspect of the character, he's also terrifying in the way he's portrayed. There are a number of scenes that display his power and his lack of restraint to use it. A personal favourite of mine is when you're running through corridors in a building on the way to find Jenova, Sephiroth's "mother", and the walls and floors all are painted with blood by Sephiroth. It puts a face of horror to the villain's mental instability to walk alongside his strength in making him a monumental bad guy.
In closing I'd like to repeat, yes I am a fan of the game, so of course what I say is more than probably a touch subjective. But all the good points aside, I never liked the materia system. It was no one thing that was wrong with it for me, neither was any problem a big one, I simply found it a general pain in the arse. I'm much a bigger fan of the Job Class system in FF5.
That's all for me, and it sure is enough.
P.S. Final Fantasy VII is to games as Stairway To Heaven is to music, and whilst I'm a fan of both, just like most guitar shops have signs above the guitars saying "No Stairway" I think game forums should impose a sign saying "No FF7". Just a thought
