Fates pretty much took awakening and improved every aspect of it from a gameplay perspective. The balancing is better, even for children, skills are better in quality and don't necessitate grinding, all weapons are involved in the weapon triangle bar beast stones and dragon stones, weapons don't break but special weapons like armor slayers are rarer and all weapons carry additional effects, there's a new debuff mechanic to combat the potential issue of stat inflation, there are status staves such as freeze and weakness, pair up was changed into two stances, each with their own perks, master seals are more limited until late game, there's a tremendous amount of variety in the level design for all routes, not just Nohr even if Nohr is the hardest difficulty, I think there's lots of creative stuff this time around.
Character wise, I think Fates characters are an improvement as well, many of them are juxtaposed with some idea or issue which make them interesting, many of them have foils and in some cases multiple foils that help to deconstruct the character a bit. The standouts are naturally the sibling characters which have quite a number of subtle touches in their characterization throughout their supports and the story (and even what they don't say). There's still some one note characters (as with any FE or really any game with this many characters [about 70]), but many are much more interesting than Awakening's cast, and they more often than not have a decent reason to join your party. The reasoning for children is quite silly though, but some of the children do provide some insight into their parents and some are pretty good as well though they aren't as solid character wise as the first gen overall. There's less repetition in supports as well.
I feel the pacing is also an improvement for all the routes, any "breaks" or transitions are at most 2 chapters long. There's nothing like the Valm arc here, but to prevent things from feeling too fast, the supports and my castle constructions help ease things out.
Story wise will be divisive I feel, from what I understand the structure is based on something called "question-heart-answer". Each of those parts is a full self contained story but focus on different aspects of an overarching story. Question is the simple set up of the world and struggle though some characters might not be fully elaborated on or merely hinted at here, the heart is an emotional take on the story which focuses on breaking down aspects of the main characters though overarching plot points aren't really tackled, the answer is the last part that wraps everything up and clears up details not covered in the other paths. In FE14, the question is Birthright, Heart is Nohr and Revelations is the answer, which is mainly why I suggest playing in that order to get the best sense of the story. Another story that uses this structure would be the visual novel Fate/Stay Night and its three routes, Fate, Unlimited Blade Works and Heaven's Feel. It's worth going into FE14's story with this in mind. There's some contrived plot devices here and there (something FE has always struggled with), but the characters are good and sell the story well, I feel.
The presentation is also an improvement I think, cutscenes even outside the CG (which there is more of than awakening, (though I think Awakening has better quality scenes by comparison because madhouse did them there instead of studio anima here) have more angles and better cinematography, the models are more detailed, they animated those models better. However, there is an issue of Pop in for battle sequences, which is a shame, though combat animations are also better.
Hopefully everyone enjoys the game though.