BF3 limited edition

I don't really know much about Battlefield, but the reason why you can't play together on the same server is probably because the map on the server only has 1 of you.

For example, Premium might have some premium maps that limited edition doesn't. Therefore, if the map on a server is a Premium, the one with the Limited edition won't be able to play it.

I'm not really sure about this, you should check it out though.

There are two items with similar names in Origin, the "Battlefield 3 Premium Edition" which includes the base game and all DLC expansion packs, and "Battlefield 3 Premium" which is a season pass for all DLC expansion packs without the base game.  If you don't already have BF3 but bought Premium you will not have a game in your "My Games" tab of the Origin client.  If this is what you purchased and you do not already have BF3 please contact EA Customer support:  https://help.ea.com/contact-us and they can fix this for you.  If you purchased the BF3 Premium Edition you will see the BF3 Limited Edition in the games section of your origin client to download.

All versions of BF3 can play together on the same servers if they are on the same platforms (PC\Xbox\or PS3)