That's a really old concept that survived from 3.5-4.0. As .NET Framework was getting bigger MS thought it would be good to split the runtime deployment into 2 separate pieces: client and full. The client profile, as it was called, was designed for desktop apps and was a subset of the full framework. The theory being if you were building desktop apps you only needed the client profile subset.
The full profile was the full framework and was needed for everything else, or if your desktop app needed stuff beyond the common core. You can read about it here.
MS stopped doing that when 4.5 was released for a variety of reasons. If you have the old client profile on your machine then you can most likely remove it (if it shows up in add/remove programs). The newer 4.8 full client has all the framework code already.