Let me go ahead and say my problems with both of them.
In Awakening, the story seems rushed. While I'm still blind to the 3DS era, I can't give criticism on gameplay, I will say that the gameplay does seem to have a lot of the same issues Sacred Stones Has: it's too easy to grind and have a capped team way earlier than intended, and this tones the difficulty down A LOT. Take a look at Blazing Sword (FE7) compared to Sacred Stones in difficulty. unless you Arena abuse through everything, FE7 is harder than FE8. This is also true for the increased stat totals for Awakening and Fates. Every unit is viable, and while this certainly isn't a problem (Pick your team as YOU want to play), it also knocks difficulty down a bit. A lot of the criticism in Awakening is due to the Avatar character, though. The game revolves around them being a main character, as opposed to them being important. This can be done fairly, but the Avatar feels too important to older players, who enjoy watching the story, as opposed to heading it.
In Fates, the story is horrible. It revolves around how important Corrin is, and every character depends on the idea that Corrin has to be flawless in every aspect. While they try to give some characterization, it's mostly a failed attempt (A good example being Peri and Xander's supports).
The final piece that I have to say is the Casual/Classic Debate; the idea of Casual Mode teaches players to play more recklessly, and it turns a lot of people away from the older Fire Emblem Games. And while newer players may complain about Elitists bitching about deaths and restarting anyways, they learn to strategize better. In a lot of older Fire Emblem games, this trial and error method felt better to the player than a simple "Oh, Raven died. I'll get him next chapter." The idea of taking away a punishment that teaches the players to play smarter, especially to the players who learned and adapted, seems unfair to new players.
In conclusion, both games aren't bad, but they are lackluster for Fire Emblem games, and that's why they get a lot of hate from "elitists." Some dislike the new mechanics, but overall, many of the ideas that made the games more popular are more punishing to newer players, and frustrating for older players.