While the PS3/PS4 release of Diablo III does sound like an improvement over the PC version, there is one downside. Those playing on consoles are stuck to their own servers, meaning there will be no cross-play between PSN and Battle.net accounts.
Community Manager Vaeflare confirmed the lack of cross-play on the official Diablo III forums. Considering the structure of Battle.net, it's not a surprise that PSN Diablo III accounts will not be able to connect with their PC counterparts. This also means that characters will not be able to be transferred from one to the other.
(Via Blue's News)