Re: but include a button for opening the page in the associated app instead.
Are you asking about the advantages for the website, or the user?
Depending on the app, there definitely are some, for both
E-commerce websites definitely don't need an app, but things like maps (set up to have a widget, directions on lock screen, etc) or media players (allowed to play in background, etc) have a use-case.
If most of the data and files are static (or rarely changing, or you want them offline) then an app is an advantage (both to you, since it loads faster & offline, and to the site since it drops data costs)
for a good app performance should be a lot better too
Plus for the website, there's some ego trip, but also having the icon on your phone's home screen is bonus advertising every time the user notices it
For most apps I actually want, there's no adverts anyway (some do it, but unless you install an alternative browser on your phone you'll still get them in the website as well)