This thread lays out all the changes:
http://boardgamegeek.com/article/7783006#7783006
The changes are somewhat significant gameplay-wise in some cases but at their core both games are very similar games. I finally beat all the campaigns in Omens and don't see a huge need to return, but I certainly have played the same bosses multiple times in Elder Sign -- there's just more variety, and you do have a better sense of stuff like monster spawns and other effects where the app just handles it for you. Slightly odd to compare, but I'd say Omens is better than base Elder Sign (and definitely better with the Omens IAPs), but Elder Sign + Streets of Arkham (where the Encounters are facedown and you only get tipped off to general difficulty by green/yellow/red color) is the best Elder Sign experience out there (and probably in my top 20 current favorites).