Best thing is still to do both. Internet is vital. This means that you cannot do majority of apps without having a web server in back end.
Basically if you build smart enough API then most important actions are still done by your web server and mobile is simply used for user interface. Mobile UI is much faster than a website UI and smoother by quite a margin, even if you use things like PhoneGap where you build mobile UI with HTML, native UI is simply better.
This is a great thing, since API's are really the way to go. Nowadays you want your awesome software to be accessible multi-platform and that means that it requires a really flexible API and UI-independency.
If you have that, then you're set