It's difficult to say. I like Awakening and Fates for different reasons, but at the same time, they have some cons.
Starting with the Avatar/Player Character, I definitely prefer Robin to Kamui/Corrin. The constant fawning over Kamui got a little irritating and I don't like how they were the main topic of some supports. Like the support between Xander and Ryoma had so much potential, but in the end they just talk about Kamui. I think that's why I like Takumi: He was wary of Kamui when they first arrived and while his attitude may be irritating, it was pretty realistic...
But I digress.
Robin gets some crap from fans because of the option you're given at the end of the game. Personally, this pales in comparison to Kamui. He's not the center of several support conversations and nobody seems to be head over heels like they were with Kamui. I also like the bond Robin has with Chrom and (potentially) Lucina.
One interesting thing about Awakening is how marrying either Chrom or Lucina has impact on the actual story. In chapter 22, the dialogue carries out differently based on your choices. In Fates, if you marry (let's say) Xander or Leo, their dialogue doesn't change when they're trying to convince the Nohr siblings that Garon is clearly off his rocker. So in a way, it feels kinda off to me.
I'm not one to harp on gameplay mechanics, so I'll make it simple. I feel Fates is more fun to play? They take away the weapon durability which takes away some challenge, but I don't know, I just find it more entertaining in combat compared to Awakening. I also like how Conquest mixes things up a bit with the maps... There's more than just "rout the enemy" or "defeat the boss".
The story of Awakening and Fates are both nice, but I think the three paths of Fates makes it less likable. Conquest, for example, was a letdown story-wise because it was like you were bullying the Hoshidans because Kamui and her siblings were too conflicted to stand up to Garon. The actions of both Kamui and (moreso) Azura in this path were also extremely questionable and downright stupid (put bluntly).
I like the time-travel mechanics of Awakening, but it seemed to start out pretty slow for me. It didn't really pick up in plot until the war between Ylisse and Plegia ended. Perhaps I felt that way because as Robin, you kinda just... get thrown in the middle of things. You don't see the war from the beginning, but you see it to the end. Fates seemed to jump right into the story before chapter 6; you get to see two sides of the coin by starting in Nohr and winding up in Hoshido by chapter 4.
On a final note, in Awakening, the children come from the future and they're their own persons. The baby Lucina is not the same Lucina from the original timeline; same goes for the other unborn children. In Fates, it seems right after you marry they jump right into having a baby then placing them in the Deeprealms to speed up their ages. I always felt that was kinda odd and I preferred the decision they made in Awakening.
I know a lot of people get squicked out when it comes to marrying My Unit with one of the "children"/second generation... But I don't think in Awakening it's as big of a deal as it is in Fates. I would think Fates would be a little more weird considering the circumstances.
Anyway, those are my opinions. I rambled more than I intended to, but oh well. Truthfully, both are great and I have a hard time choosing because I like some things in Fates that I dislike in Awakening and vice versa.