I play only live. I've tried both. I find playing a game without databases and in a sitting to better simulate OTB chess. I play games of a at least 15.
For me, online chess breaks up the continuity of a game. I could certainly take notes and write down plans, etc. But all the same there is a certain adrenaline rush which comes from playing an OTB game which is present in live games and absent in online.
I also feel live accurately tests my tactics, plans, etc. within time constraints. If I want to research and study, I will analyze my games or work with a book.
I have played correspondence ("Online") continuously since 1996.* My OTB rating has risen from mid-1400s to 1982 (recently dropped to 1899) during that time. I have also played an estimated 75,000 live games online during that time.
Most of my live has been three to fives minutes total per player (no increment), but I have played a lot of different time controls. The quantity of blitz and bullet has given me experience with nearly every conceivable opening and endgame, but has cultivated intuitive responses and reckless attacks at the expense of careful calculation.
Correspondence chess has cultivated deep positional understanding, forced complex and accurate calculation, and has aided my development of new openings through focused practice against formidable opposition.
Probably the bulk of my personal OTB improvement stems from study rather than play, but everything contributes. Live chess is a mixed bag: it has both benefits and negative consequences. Correspondence is wholly useful, and it motivates extensive and deep opening preparation with an eye towards understanding middlegame planning.
*I've gone from postcards to email to sites like Chess.com. I started playing "turn-based" (online correspondence-paced chess) in 2003.