The newest research which I found is a statistic from september 2012 by facebook itself on this blog: http://www.forbes.com/sites/benedictevans/2013/01/02/facebooks-470m-mobile-app-users/
It says, the facebook app counts 470m users. The interesting sentence is the following:
Out of 1.07bn total users, Facebook reported 604m ‘mobile users’, implying that 134m were not using these apps and hence were using the mobile web
So of all 600m mobile users, only 130m are using the web, the others using the app.
But september 2012 is a long time ago. So I search for newer statistics. The only one I found is not directly about the using of the facebook-app itself, it's about time spending on apps vs. web.
You can see, 86% of the time the user is using his smartphone, he is using apps. And 17% of these 86%, they're using the facebook app and not the web.
After all I would say, about ~78% of the users are using the app instead of the web (2012), and if I look at the image, the percentage of app-users are increasing.
BTW: I also use the mobile app.