Apps will give you a few more obscure datapoints that web analytics won't but I'll bet that very few companies actually use them. I doubt many even study bounce rates when they present you with these app download modals.
I was thinking more device orientation/velocity/camera etc Geo is available on web, and SSO via fb etc allow virality through contact lists in much the same way as an app (I think). I guess the biggest reason why a business should get you to install an app is push messaging (though I guess that could be approximated with email).
An app with access to your camera can shoot fucking video or pictures and run them through analysis. For example, an internet radio station can adjust its playlist based on who it sees. Well, that and its ads.
What internet radio station is "access[ing] your camera[,] shoot[ing] fucking video or pictures and run[ing] them through analysis" to "adjust its playlist" and ads "based on who it sees"? For that matter, what app is doing anything remotely close to this?
Of course there are things you can do with apps that you can't do with sites (though I guess HTML5 is looking to challenge this) - but the OP is talking about sites pushing app versions of themselves through modal blockers to the content she/he is trying to reach, not separate apps which couldn't be remotely realised as sites.