PC version is stored on battle net. Console is stored locally hence character transfers from last gen to current gen consoles. It's also the reason console version is not getting seasons/leaderboards.
BTW, I consider this game's gameplay simply legendary. It's so smooth, and there so many items now that are game changers... you can play for hundreds of hours and still discover new ways to play a class, not to say if you play different classes...
I just love this game. It's incredible. Love it.
For those who played the first console version, gameplay design has changed so radically from then to the current game (post Reaper of Souls expansion) that the game feels completely new. Not joking here, I was surprised back then. So don't worry about the game feeling old news, you're gonna enjoy it a lot anyway.