1. Dragon Age has better-developed, more memorable, and overall funnier characters than Mass Effect.
2. The overall storyline of the Mass Effect Trilogy feels like a typical science fiction "save the universe" trope. Dragon Age, on the other hand, is less of a trilogy with one protagonist and more the story of many people and their interactions with each other over a period of great turmoil.
3. Dragon Age's morality system is much more ambiguous than Mass Effect's, making it more difficult to decide the best course of action one should take to affect the story.
4. While Mass Effect's combat system is based around shooting, Dragon Age's focuses on more archaic weaponry such as swords and bows, giving Dragon Age the edge in terms of strategic and epic combat.
5. The Mass Effect Trilogy had a weak ending and a strong middle. In contrast, Dragon Age's middle was weak, but it ended strongly. In my opinion, it is more important for a series to be strong at the end than at the middle.
I encourage all commenters who agree with me to share other reasons and those who disagree to offer counter arguments.