I’m going to guess as I don’t know for sure… with the app, once you’ve created your profile / proved your identity etc once, the first time you use it… Any updates (such as future boosters or whatever) can be automatically added to your profile and you can then access your latest status really easily. I suppose they can also add things like info on test results as well, to get a complete picture of your Covid history in one place. In other words, for the user with the app, any updates are added passively.
With the download on the other hand, the user has to actively request an update (‘send me a new one as I’ve just had a booster’), and it’s more of a faff as each time you have to go in and prove your identity. I suppose the same principle as the app could be achieved with a ‘user account’ on an NHS website, but an app is probably more likely to get higher usage, as smartphone users are very used to the idea.
That’s my guess anyway.