Ah, ok, the normal stuff then. Good to hear, since my controller is out for delivery!
The issue comes from two points, really. The first is that different controllers can have varying amounts of input lag. This shouldn't be a massive problem unless the manufacturer screwed up, but it can cause issues when switching hardware with games that have very tight timing requirements, and Project Diva is going to be tighter than most games (really anything outside of Beatmania or fighting games with no button buffering). The other is that Project Diva: Future Tone
does not have any input latency adjustment. The lag adjustment in the game only works for audio, so you really are going to want to play this game on a TV or monitor with very low latency.
Really nitty, technical, and probably needless sidebar: If I had to guess why, it's probably due to how the scoring works, specifically with holds. There are some songs that end with a hold where the song ends before you can get a max hold bonus, and allowing people to move the timing window earlier relative to the song would mean that you could squeeze out more points from that final hold (as the holds award points on a per-frame basis). Granted, you could then push back the end of the song with the sync adjustment, but then you might break other songs (and this is an actual bug in Project Diva F/F2 -- if you have too extreme of a timing adjustment, some technical zones become impossible because they end before you can hit the last note). This could be addressed in development by setting some style guidelines (like not having songs end on a hold that can't hit max bonus and having at least x ms of dead space at the end of a song (x being whatever the max offset it). That would have either required some real forethought (maybe too much) or deciding that arcade/console charts don't need to be 100% identical (which may have been worth it, especially after the scoring change).
This is probably getting more into the weeds than almost anyone would care, though. :x For most people, they're just going to see hundreds of modules and songs and they're just going to want to push buttons with Miku -- and they'll be right.