Not a fan of either.
I enjoyed Awakening initially, then came to notice how terrible the story is and that I didn't feel attached to any of the characters like I was for characters in the older games. I dislike time travel in general, and the DLC stuff made way for the terrible and lazy Outworld concept (which I might add was never a thing in the series before Awakening). The map design is terrible in that most of the battles take place on giant plains and it lacks the varied objectives of past games (Survive/Defend, Escape, Seize).
It encouraged new fans not to bother to learn how to strategize in a strategy game and accustomed them to endless grinding, which wasn't a thing outside of Sacred Stones. It was the first impression of the series for many people, and it is terrible at even doing that since it is so removed from the series standards.
Meanwhile, I couldn't bring myself to spend $60-$80 on Fates because too many things seemed too far flung from what I used to love about Fire Emblem. I don't like shipping, they took out the series staple weapon durability, and the way they explain the child characters is ludicrous. Fates just comes across as annoying pandering.
I was originally interested in at least picking up Conquest because it was supposed to bring back the strategy that was sorely missing in Awakening, but from most accounts the story is godawful. For me to get into these games, the story has to be alright and the strategy has to be solid. If either of these things is lacking then it just saps me of any incentive or interest to play it.
In the end, if I absolutely had to choose, it would come down to whether I place more value in Awakening's weapon durability or Conquest's varied objectives. In that case I lean toward Awakening because even though it's lacking in objective variety and has nothing of any real appeal to me, it was at least somewhat related to the series I used to like.