Please don't hate me, but they technically are different games, not just different versions.
From Siliconera: "Fire Emblem If: Black Kingdom has a more complicated storyline and gameplay. Players will have to deal with limited resources as they try to revolutionize the unruly Nohr kingdom from the inside." "Fire Emblem If: White Kingdom is a more traditional Fire Emblem game where you side with the Hoshido and players will be able to obtain experience points plus money outside of the core story." Source: http://www.siliconera.com/2015/04/02/fire-emblem-screenshots-takes-us-inside-black-kingdom-white-kingdom/#d7b3QGq38iQfErXK.99
Black Kingdom sounds more traditional to the Fire Emblem system, but White sounds more like a casual, traditional JSRPG experience. They will both be different sides with different features and different experiences. Instead of singular games with multiple sides but the only differences is dialogue, boss fights, endings, etc. (like the SMT series, Bioware RPGs, and the inFamous series), or games that come out with multiple versions at once but the only difference between them is color scheme/theme, obtainable things, etc. (like Pokemon, Mega Man Battle Network/Star Force), these games sound like they will have different rules, tasks, systems, choices, etc. Essentially, the difference between them sounds like the difference between Final Fantasy VII & VIII, Fire Emblem 12 & 13, Persona 3 & 4, etc. What I'm saying is: at their core they're the same, but there are plenty of layers of differences on their outsides.
The fact that it came in two just shows how ambitious this project was and shows how what IS where trying to make couldn't fit in one package because of technological limitations. Instead of costing us a new console to make this a New 3DS game, it is going to be a divided experience in which the developers are offering us a generous discount if we choose to invest in them both. I actually feel thankful that they put this much effort into a game from a series they were considering cancelling.