A lot of people are struggling with the choice between a mobile app and a responsive website.
When you look at the discussion of responsive vs app, the answers have been given already. It basically all boiles down to whether your content is static or interactive. Static content (like posts on a blog) has more use on the web, where interactive content (like most webapps, Google Docs for example) fits better in an app at the moment.
A pattern I also see more often nowadays is to tell your mobile web users to download the app from their store on a responsive portal page. This is a great way to send users to the correct location if you don't have a responsive website. At least it's better than annoying or even losing the user by giving it a desktop site that's way too interactive.
Currently we know nothing of app store search statistics, which is weird since web search statistics are publicly available. In the case of Google's web search there are tools available which give you considerable insight in web search behavior. For app stores, there is no such thing, Google and Apple aren't giving away app store search statistics, which really is a shame. I hope eventually this will be made available for the app stores as well, but for now we're fucked.
The most common reason for not building an app for your interactive website, is lack of budget of course. Well then don't do it yourself, but let the world of bored developers get creative! If you make an API and some simple documentation available, everybody can build an app with it! You can start simple, and extend it with more functionality over time if you want/have to take it real easy on budget. An API will generally cost you way less than a fully fledged app. And you can immediately start listening to your users and other developers. And believe me: you'll want to build your own app pretty soon. It won't take anyone long to see how much profit they're missing.
As an interesting side-note; a while ago I suspected facebook of doing something smart, but it turned out I was wrong. I had hoped they created a small app that is essentially a stripped-down browser with no options other than showing the already awesome app-like responsive website. As far as I could see, their app matched completely with their responsive website, except for the login page. Later, when I started looking more closely at the differences in layout, I do think the app is native, and not just a browser. But still the idea would not be that bad would it? At the very least it's another cheap way of creating an app.
The way I see it, when you have an interactive website, you'll simply need both a responsive site, and an app. If you don't, you'll lose users/visitors and thus possibly money. You can however temporarily get away cheap and easy: with a portal page for mobile users, building a API, or doing what I thought facebook did. But the real solution is that app of course.