These are two [three, see my last paragraph] different but very similar-sounding effects - the modern one, as demonstrated in the OP, is Autotune. It does real-time retuning of singing or talking (or any single-pitched sound) to the nearest note on the musical scale it's set for (could be chromatic, but is usually the scale of the key the song is in). It in itself often does make a voice sound robotic (it was originally meant to fix flubbed singing notes, and to be adjusted so it doesn't switch instantly between notes, sounding more like a natural singing voice), but mixing in a bit of the original vocal causes the characteristic 'phasing' sound that the vocoder doesn't have.
The earlier one is a vocoder, and wasn't nearly as often used as Autotune has been in the last decade. A vocoder uses TWO signals - one, usually the voice, is a "modulator" - it dynamically sets filters to the same spectrum as the modulating signal. The other is an instrument, usually a "standard" music synthesizer, goes through the filters, thus the spectrum of the vocal is impressed on it. This is obviously what Herbie Hancock was using in that clip, "singing" and playing the synth at the same time. The pitch of the voice is ignored, and it uses the pitch of the synthesizer, as one can hear when the vibrato is turned up.
One unusual use of a vocoder is on Pink Floyd's "Animals" where dog barks are the "modulator" and the carrier is a Hammond organ - the output is short "barks" of organ sound.
Perhaps the first use of a vocoder on a recording was by Wendy Carlos for the "Clockwork Orange" soundtrack. She talks about the vocoder and its whole history here:
http://www.wendycarlos.com/vocoders.html
I don't know of an online recording, but the vocoder can apparently be heard on this CD, and IIRC that part didn't make it to the film:
http://www.wendycarlos.com/+wcco.html
I think I've got the LP somewhere, but I've heard an outtake on Carlos' "Secrets of Synthesis." CD.
Autotune really came onto its own with the Cher song, though the guy who mixed and engineered it swears he used a vocoder for the effect, and he may well have, but many people suspected he was also using Autotune to "tune up" Cher's not-quite-perfect-note singing (as it had already been used on many top hits), and that's where it all started. I recall whatever Kid Rock song it was not long after that which used Autotune heavily, IIRC it was "
Only God Knows Why."
Autotune had already been in use for a couple of years for its originally intended use, and a careful listening of a song where it's been so used can often find a slight touch of "that sound" in the vocal. More and more songs over the past decade+ have been using it for singing note correction, rather than the previous technique of having the vocalist sing a song several times and the recording engineer picking the best track for each line, measure or phrase during mixing.
One more thing, the video in the OP has "voice box" in the title - this was an electro-acoustical device sort of equivalent to a vocoder, I think more popularly called a
"talk box" It takes an instrument sound (usually electric guitar), puts it through a speaker and into a hollow tube that goes to the singer's mouth - this allows the mouth to "articulate" the instrument sound with his mouth as if he were singing, but the sound comes from the tube and the instrument rather than his/her vocal chords. The first use of this I recall is Joe Walsh's "
Rocky Mountain Way," about three minutes in. It was especially prominent on Peter Frampton's huge hit "
Do you Feel Like I Do?" from the live album "Frampton Comes Alive."
And, of course, over three decades later, Peter Frampton
is still doing it.
