>There is no way for an app to install on your profile without your permission

How I understand the concept of the like button is this:

The like button is an extremely easy version of Facebook connect. When you are logged into Facebook and visit a site with the like button implementation, you have automatically logged in. You do not need to click the like button. (Remember you could once sign in automatically to TechCrunch's commenting system by just visiting).

This way Facebook can monitor the sites you visit and in future can serve you better ads on those websites than Google. (that is how they are a major Google competitor.

The so called apps that were installed without permission was was just a internal tracking tool that was made public. It still exsits but users just don't see it any more.