Overall, I love it. I recognize that it has to have some limitations on mobiles and while part of me wishes it didn't, the good far outweighs the "Aww, I wish it had X!" moments.
I use a desktop (currently hooked up to my TV) at home, and have a tablet for anything it's easier to use for, including travel. Although the tablet only works on Wi Fi, it feels like my access has doubled with the new app.
Limitations: As near as I can tell, in the app you cannot read mail, see other players' cities (or their city names containing help preferences), craft spells, cast spells, work with ancient wonders, participate in tournaments, move forward in special events like the Halloween one (although your actions will count towards it) or build or use buildings/summonings won in special events. If there have to be limitations, I think these are good choices as most are things you only need to access occasionally.
Strengths: In the app you can build, move, sell, manufacture, harvest (coins, spells, supplies and goods), research, train, use the trader, handle visits to both fellows and neighbors, view notifications, view and complete regular quests, invite and accept new fellowship members, chat with your fellowship, and resolve regular encounters via the world map. And all of it is fabulous. All of these functions seem easier in the app.
For me personally, it has three really big benefits.
1. Visibility
I have eye issues, nothing new, but quite noticeable recently. The eye issues mean that the setup with my desktop on the TV isn't ideal; I usually have to blow text up to read it, due to the distance. When I can't, I have to drag a stool next to the TV and sit close enough to read it, which I can only do for so long because the stool gets painful to sit on after a while. The app allows me to see chat without doing that, because the tablet is in my hands and as close as I need it to be. Mail isn't viewable in the app, but it's less to need the stool for, so the app is a big benefit here.
2. Other accessibility
This has made the game far easier to play. I have been prone to tendonitis this year, so every time I have to use a mouse or trackpad to hunt for things (the best culture in someone's city to click on, for example) it can cause pain. I use different motions on the tablet than using even a trackpad, and that usually makes the difference in avoiding pain. So that's a plus, but the other plus is in how visits are handled by the app. It provides one culture location to click on, without the hunting, and from the choices offered it looks like it's offering us the best choice currently available, which is doubly awesome!
3. Travel
For travel, it is going to be perfect. Prior to this I had a browser on the tablet that could sort of pull up Elvenar but hung a lot and made it very difficult to get anything done. With the app, I can get into the game and do what I need to as long as there's any decent signal.
Summary: It's a very good tool and should enable an increase in activity for some of us, but it won't entirely replace the web version for most active players.