For shooters, monitor... not even close. Not one TV out there that can match the response time.
For sports... this is a tough one for me. TV helps me see more of the court on 2k, but the monitor allows me to use the shooting meter more and not worry about input lag as much. I shoot better on my monitor, but pass better on my TV.
Games like Zelda? Better on TV.
So I think it depends on what type of game, the game mode and if input lag plays a big role in your success.
Also depends on the person.... with muscle memory, you can use the TV and not have any or little effect.
I started using a monitor for COD, about 3 years ago. Never looked back. I can't use anything else. I used to play competitive Counter-Strike, back in the days and was already used to playing on a monitor.