I was sitting at around 60% done with the campaign. I had jumped into it right after finishing Birthright in the first week that Fates came out and just gave myself burnout on the game by playing so much. Now, coming back to it, it only took me another two days to finish off the campaign and wrap it up.
It's really cool to see the other side of the story. Birthright covers siding with the Hoshido side and Conquest is about picking the Nohr side. This is definitely the darker story. Hoshido is more conventional and you get to be the "good guy" pretty clearly. The Nohr side sees you trying to change things from within an evil kingdom even though you're trying your best to not be evil. It's actually more interesting because of that.
But, it's also the harder campaign by far. Birthright has some challenges but most levels see you trying to clear the field or defeat the boss. Conquest has much more varied objectives and it's better for it. But, some of them are super difficult to actually pull off. I had fun with it, but this would be a bad place for a Fire Emblem newbie to start.
If you're thinking about grabbing Fire Emblem Fates and you never have played a Fire Emblem game before then definitely get Birthright. If you're a Fire Emblem veteran and want a challenge then grab Conquest.
As for me, I'm probably going to pick up the Revelations campaign soon and play my way through to see the final "true" ending. I just have to be careful to not immediately jump into it and give myself burn out on the game again. Because, honestly, this is one of the best games on the 3DS.