You've also "played" each class by controlling your party members. Whether you're the Mage or you use Morrigan or Wynne, the end result is you've chosen different spells and used them in battle. The same goes for Rogue and Warrior types.
I would recommend playing through all the character's origin stories, however. They're very different and quite interesting. I've recently completed the City Elf origin and I really liked it. I've read that the City Elf origin is actually better if you're female, which makes sense considering the events that take place.
Probably the most important story-related differences are those that have little-to-nothing to do with your character's origin or class. There are some minor differences in the story based upon your origin, but those don't affect the entire game as a whole.
I'd suggest playing DA:A (make sure to import your character from DA:O) and play that. I didn't feel DA:A was quite as good as DA:O, but DA:A is an expansion and not a full game.
If you're tired of DA and a fantasy setting, I'd say go buy Mass Effect and Mass Effect 2. Those games are great (ME2 is better in my opinion) and take place in a purely sci-fi setting, which is unusual for RPGs.
The ME series is more action-based in that you mostly control your character and issue orders in real-time during battles. You CAN pause ME1 and ME2 (which brings up a HUD-like screen and interface) to change weapons, use skills/powers and tell teammates where to go, but it's an entirely different style than DA.
In the ME series, your character is fully voiced instead of remaining oddly silent, like in DA. Plus there are a lot of character creation options and classes with different abilities to choose from.
Go watch some videos on YouTube to check it out if you're interested.
