Assuming you play on closed battle.net (the "real" battle.net) then yes, they are separate. You should be able to play on open battle.net with the CD Key you already have.
Open bnet characters can be edited/hacked/ridiculous stuff. You know, characters that do 1000x the amount of damage they should do. Kinda pointless in my opinion, killing every boss/character in the game with 1 hit...
Closed bnet has the ladder and such and is officially regulated by Blizzard. It's identical to the single player game except you can have more players. Most people don't play SP anymore as you can play the entire SP campaign on closed bnet by yourself or with people.