The Ancient Knowledge Point Instants (AWKP) can be used in your Ancient Wonders on the mobile app. So, if you've built Golden Abyss or Tome of Secrets or any other in the AW build menu, you can insert the instants there to fill the research bar so you can upgrade it and improve the benefits. Regular kp earned over time in the game or from tournaments or purchased or whatever can also be used there. AWKP cannot be used in the research tree to unlock techs; they take only the regular kp.
The confusion with device types is related. In the browser version, you can also donate both regular kp and AWKP to other players' AW's in the same world as your city. Top contributors get prizes (runes for that AW and/or AWKP) when the research bar is filled. You can't earn rewards for contributing to your own AW's. Fellowships often have various AW swap clubs (hosted in the in-game message system that is also not yet accessible in the mobile app) where you donate to other members' AWs and they donate to yours so you can get the rewards. The mechanism for doing this on browser actually allows you to click on any city on the world map (other places, too, like FS member list, notifications, etc) and actually go in and look at the city. While there, you click on an AW and it opens the appropriate windows where you make donations. Unfortunately, this is also how you give neighborly help on browser. There are 'hands' to click on and then you click on the building to help it. You have to go into each city to give help. Takes for.ev.er. to do visits on browser. Also, cycling Fellowship Adventure quests for badges is much quicker and less mistake prone on the app rather than the browser. Setting productions is quicker on browser with hot keys.
From the last I heard of time frames (which may have been last winter/spring?) the priority was to have all the game features available in the mobile app by the end of 2018. They're also testing HTML5 in the browsers on beta as part of the plan to migrate away from Flash that runs the existing game on live browsers. I don't know if there have been progress reports, but I've seen steady feature releases on the app with just about every mobile update. My guess: as both these developments occur, the game will be similar on both platforms.What that ends up actually looking like? Who knows, lol?!