Amazon Prime
Amazon do quite well with ensuring these is a similar experience between TV and app. But that can actually cause a few issues. When I open the app on my tablet I have to choose a profile, on my tablet, used only by me... They aren't alone with this, but you should be able to go straight in without having to choose.
One area I've always thought needs improvement is the watchlist on the app. On a phone it is probably ok, but I use my tablet, and just having a list feels really awkward, and it isn't a great experience having to scroll though it.
We also have a secondary menu at the top for movies,TV, sport and live. But you cannot easily go from one to another, it's a pitfall of having a side and top menu - so again it isn't actually ideal.