For the competitive scene you will want to play MW. Cold war in my opinion is more of a casual kick back with your friends kind of game and maybe enjoy some zombies while you're at it! Haven't played the campaign but multiplayer in my opinion is very different from MW. You move slower, takes a bit to get used to the feel of the time to kill and for me personally, when I play this game for a bit and switch to MW I appreciate the mechanics of MW a whole lot more just for the fact of the movement feels seamless and the use of the way too overpowered guns.
Btw sniping in cold war after you get used to it is very satisfying.