Battelog was free an you had to sign up for it when you bought the game to get access to multiplayer. That has since changed, but when I got it that is what you had to do. Also you can only change your tag on battlelog so you have to log on to make in game changes. Ironically you can not change your loadouts from battlelog like you can in Elite.
Premium was release AFTER the first DLC was sold, so if you bought Back to Karkand for full price (Like I did) you do not save as much as if you waited and bought Back to Karkand as part of the Premium Package. I think EA is a douchebag asshat for milking us like that, but I understand that this is par for the course for them. *shakes fist at the sky where EA lives*
Battleog is great I think. Fully functional from day 1 with no problems! But the forum is one of the worst things I have read. The content is good, format is fine, but the people.. Dear god the people on that forum are insufferable. There are more trolls per square inch than anywhere I have ever posted.