The biggest difference is that for real cross-platform support you need to be careful which APIs you choose to use; some have cross platform equivalents, but not all. In addition, we didn't want to just render the iOS UI on Android, so we replaced some of the components, about 10% or so to get a more native Android feel. Because the iOS version is more mature, we started with it and migrated it to Android a few weeks ago.
The biggest thing, and this is not even React Native specific, is to ensure that the components are implemented with flexbox correctly and the layout adapts properly between screen sizes. Because Android has such a disparate set of screen sizes and devices (and iOS lets you "cheat" a bit because the set is so much smaller, letting you hardcode sizes/layout that you shouldn't), some discipline is needed.
Lastly, your top-level container component will likely use padding on iOS to not overlap the status bar, don't forget to remove that on Android, because it renders it the way you'd expect: below the status bar, not underneath!
