I use my iPhone for 99% of all my internet browsing. With only a few exceptions do I opt to go to the "Full Website". Most aren't optimised for mobiles but with the screen capabilities if the iPhone that isn't really a concern for me, pinch and zoom has almost become second nature.
As for apps. I would use an app over the web in most circumstances as they are more compact to use, use less data to maintain, are "trimmed down" to all the features that I use most and I can turn off the background refresh feature.
Security, that's a big grey area. For example, Facebook wants access to your contact list before you can post videos via the app. Geo location too can't be turned off too for all apps that use it.
I have been using an iPhone for 6 years as my primary gateway to the internet and only a hand full of times it has been frustrating.