Two Sides of the Story
Fire Emblem Fates is definitely a bit of an experiment for the series, and has a unique structure. There’s two different versions of the game, but the differences between the two are rooted much deeper than something like Pokemon X and Y.
Birthright and Conquest represent two sides of the same conflict. The world in Fire Emblem Fates is separated between the two kingdoms of Hoshido and Nohr, royal lines descended from dragons who each worship different deities. The player’s avatar, named Corrin by default, was born in Hoshido but abducted at a young age by Nohr.
Your choice on which version to get decides which side of the conflict Corrin lands on. One version of Fates runs $39.99, and whichever one you don’t purchase can be downloaded as an add-on later for $19.99. There are major differences between the versions in story and gameplay.
Each game shares the same base for their structure, and the first six chapters of the game are the same. Once you make a choice between the two though, the subsequent chapters until the end of each game are very different. The core gameplay remains the same across both, but the difficulty of battles and the overall experience vary between Birthright and Conquest.
Your decision on which version of Fire Emblem Fates could be decided by a few factors, particularly your experience with the series and tactical strategy games in general.