If you need a presence on mobile platforms there are 2 ways to go: mobile native app or a mobile-tailored site (or both). This kind of choice isn't new, it's as old as the web, either make a native (fat) desktop client or make the browser the client (or both). But the rise of mobile puts a new slant on it.
The way things stand now a mobile app is generally superior because of it's nice tight integration with the native UI and all the features of the hardware, but there are indications that the capabilities of mobile sites will someday minimize that gap.
I agree, The world is now becoming smart with the smartphone development. So, that's true that having a mobile app is more beneficial because it is developed in keeping mind the mobile , hence it is more UI friendly.
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.