What's the difference between a vocoder and a harmonizer?

Question

As a follow-up to this question: https://video.stackexchange.com/questions/351/what-is-the-difference-between-autotune-and-a-vocoder, let's get into the nitty-gritty about vocoders and harmonizers.

For example, in Imogen Heap's famous Hide and Seek song, she's apparently using a harmonizer, not a vocoder. I also read that the harmonizer as a live performance device can be very unpredictable - to the point that Imogen won't even use it in a live performance.

So how does the harmonizer differ from the vocoder, where you can just play the keys and get the source tuned to the carrier frequency?

Answer 1

"...from the vocoder, where you can just play the keys and get the source tuned to the carrier frequency?"

That's not what a vocoder actually does. A vocoder analyses the rough frequency spectrum of a (source) signal, and uses this spectrum as a kind of equalizer envelope for the processing of another (carrier) signal. If the carrier had a very even frequency spectrum before, it will thereby pretty much take over/copy the frequency distribution of the source. The source itself is not processed in any way at all, apart from being analysed.
The carrier is usually such an even-spectrum synthesizer sound, but you can use any kind of signal. For instance, you can use a string quartet and modulate it by feeding a drum set as the source signal!

On the other hand, "where you can just play the keys and get the source tuned to the carrier frequency" describes quite accurately how a harmonizer works. The problem with this is that you need to know the exact frequency of the source. It needs to be a proper clean monophonic sound, otherwise the algorithm can't know how much it needs to pitch-shift the signal to reach each of the carrier frequencies.
What's more, in a harmonizer you usually don't specify the target frequencies manually, but just pre-define some scale for the harmonizer to work on. It then needs to figure out by itself which harmonies to actually create, which again only works if you start with a very clean signal.