Mobile app or responsive mobile website . where do we need to invest our efforts?

I vote for responsive websites:

1. Cross-platform !!! One design runs through ALL platforms.
2. Easier development and deployment cycle.
3. Easier maintenance.
4. Cheaper; it would take me2 years to explain how it is cheaper!
5. More ads to your service: More money to your account.

For whom going to tell you that websites require internet connection; I say whom is not subscribed to an internet service packages?

Your targeted segment are people with smart phones, rarely you find someone who is not online!!

Regards,

An app is still misunderstood, if you need to have the same info from the website and host them in Playstores then create a mobile site and converted to an app since it will be easier to maintain while if you are seeking notifications, push notification and a system that stays running even in the background with GPS, camera options and many additional extended features that will use the mobile hardware rather than your server's resources (bandwidth and speed) develop an APP.

If you need to access the native features and want to get closer with low level of mobile operating system, than native apps is your best bet. Otherwise you should go the web version as it was best explained by @ maisoon al quraa. For instance, game development need to be designed using native languages. 

Web View is the bridge between your app and the underlying OS, develop accordingly.

i prefer the responsive website, at least the HTML5 + CSS3 include the @media can build mobile app through several convertor like adobe air and more...

but its going up to what you need to be...

Responsive Mobile Web OR Hybrid app in something like Phonegap. Choice depends on features your application has that may or may not need to use mobile sensors, user base and how they plan to use the application which might require offline capabilities and several other factors..