Synthesizer is the part that creates the sound, keyboard is the part that tells synthesizer what note to play, when, for how long, and possibly at what volume.

Synthesizer is the part that creates the sound, keyboard is the part that tells synthesizer what note to play, when, for how long, and possibly at what volume. Sometimes people call the whole thing keyboard cause its the most distinctive part of it, but you can use the same keyboard connected to different synthesizers and synthesizers can exist without keyboards. What's important about synthesizers is the fact that for the first time in history it allowed musicians to control the "character" of sound gradually in new dimensions (the parameters that let you distinguish the same note played on piano, flute, guitar, violin, etc.), creating sounds that were impossible previously and even changing the character of the sound in real-time as another dimension of artistic expression. It's like you played a long note on violin and it morphed slowly over time into a flute and then some instrument that doesn't exist. You couldn't do it before and all these new possibilities and constraints changed music. Also the particular UI of some synthesizers allowed easy exploration of these new dimensions and that's important too. It's one thing to be able to play any waveform you want (you can do that by editing .wav file in hex editor), it's another to have several knobs and sliders and hear the differences in real-time when you tinker with them. We can simulate all of this in software but not 100% perfectly.