The basic difference between a synthesizer (or a MIDI keyboard - because it produces MIDI protocols) and a keyboard is the fact that synths are used for creating your own sound or adjusting different samples in very great detail. On the other hand, keyboards have been designed to offer people that want to play them a large variety of sounds and samples that are all automated.
Furthermore, keyboards are meant to reproduce the sound of a piano first of all, and after that take care of things like reproducing other kinds of instruments. Essentially, the keyboard is a proper instrument, while the MIDI keyboard is a controller that you use to produce music in a studio.
Of course, you can still play the synthesizer, but because it’s not made specifically for that purpose, it won’t feel so good as a keyboard. Furthermore, the keys are often limited in number when it comes to synthesizers.