The mobile app has finally almost caught up with the browser version now that you can put KP into other players' Wonders. There are still a couple of things that make either the app or the browser version better, like how easy it is to visit map neighbors with the app, or how the app version of the Trader is a steaming pile of crap compared to the browser version. I never use the Trader when I am playing in the app. Instead, I always wait til I can use the browser version when I need to make or take a bunch of trades. This is especially true once you get to chapter 12 and beyond and the sentient goods start getting added. I am sure others will point out some things I am forgetting about, but those two are the big differences that I can think of right now.
Edit: one other big difference is when you fight in provinces or Spire or tournament with the browser version, you can choose between manually fighting the battles or letting the game auto-fight for you, while with the app, you can only use the auto-fight option. The browser version also gives you more info on your troops than the app.