Let's be very precise about what we're talking about: A MIDI
controller is anything that can send MIDI commands to another
MIDI device. Keyboard type MIDI controllers are only one kind.
But, for the purposes of this answer, let's assume that whenever
we say "MIDI controller" we mean the piano keyboard type. A MIDI
sound module is a device which accepts MIDI commands, and
produces a sound. Those sounds could be artificial sounding
beeps, realistic sounding simulations of a real instrument, or
drums. Let's forget about the other types of sounds, and assume
you're interested only in piano sounds. What you refer to as a
"keyboard", I will call a digital piano. It's the MIDI
controller, and the MIDI sound module, all in one unit. (It's
possible that internally, it's not using MIDI, but that's a
detail that's not important). It's common for these to also have
MIDI connections, so you can use them standalone, or as a
controller, or as a sound module. MIDI commands can be carried
over traditional 5-pin MIDI cables, b