PHP frameworks have really taken up some new concepts from other technology such as NodeJS and Ruby. On my search for an easy deployable framework to server, I found both CakePHP and Slim are good enough.
For getting started with development CakePHP seems like old school as one just needs to download the package and put it on a server and get into coding. While Slim seems to be a little more learning as one needs to get used to Composer. CakePHP also has an option to get started using Composer.
CakePHP is a full framework having everything to get an application running on server. Slim seems to be a better option to get upto speed with API development.