Well for starters, in Crisis Core alone there are a couple things. How does Cloud go from being able to one hand the buster sword when attacking Sephiroth to not being able to even carry it at all when walking at the end.? Sure, something simple and will probably be answered with "Well he was weakened." but it's still stupid.
Another thing, not so much continuity, but going from Crisis Core to VII, it really makes Cloud seem like a real dickhead. Yes, Zack told Cloud to live for both of them, but he didn't tell Cloud to take over his life. In FFVII, we are made to believe it was some sort of mako or memory issue on why Cloud was pretending to be someone he wasn't. From the events of Crisis Core, he knew exactly what he was doing and just dicked over Zack and stole his life. Hell, he never even told Aeris that Zack said hi, which was one of the last things Zack said.
Then there is the whole Sephiroth issue that goes through all 3. In Crisis Core, we are to assume he is basically a clone/child of Jenova. To me, this makes me believe he is a test tube baby and has all the powers that Jenova has. In FFVII, sure, same thing. He is the child of Jenova, test tube baby made by Hojo using Hojo's own sperm to make the baby from Jenova's egg or however they do it. Then we get into Advent Children, and the "Brothers" are all potential Sephiroths. How the hell can they just turn into a person that was a clone? I understand the whole Jenova influence, about how they were all related in that way, but how can the strongest of them just *poof* into Jenova? Why didn't Cloud turn into Jenova? It's basically explained that the strongest brother would be turned into Jenova, and wouldn't that be Cloud?
Then there is the whole curing abilities of the lifestream. At the end of Crisis Core, I am lead to believe that Genesis was cured of his degradation by the lifestream. How come the life stream didn't do that for Sephiroth the times he was in contact with it? Sure, he wasn't degrading, but the "monster" in him should have been destroyed. Instead, Sephiroth just sits in a bubble above the crater basically dead, but not cured. I just don't get it.
Thats it for now that I can remember, I am sure I will think of something else.