First lets take on
The big difference between the two can be expressed with these two keywords:
程度 (you could think of it as "degree" or "extent") and
頻度 (frequency) (credit @sawa for pointing out ほとんど has both)
頻度 only (or
回数の多さ, you could think of it as "frequency" or "how often something happens (or is performed)).
めった in 漢字 is
滅多. Notice the relationship with
Lets look at some examples:
In the above examples, the
ほとんど cannot be replaced with
めったに. Do you see why? The concept of "how often" for the ability to read
漢字 and ability to hear someones voice doesn't make much sense.
However, lets look at an example were both can be used.
Both can be used here. However, I feel that
ほとんど～ない is more common (Googling both phrases gave me a lot more hits for
テレビをほとんど見ない.) There also may be other subtle differences, however I could not find conclusive evidence at this time (if anyone has any insight please let me know!).
When I read the above (the action of seeing), I think
めったに～ない is better here (unfortunately that might be a personal opinion).
Now, lets add
まれ to the bunch.
まれ falls under the same category as
めったに, in that it expresses frequency. The difference is in the usage (the meaning is exactly the same as
まれに見ない // Not Japanese
まれに見る // Okay
まれ technically is already a negative expression meaning
実現・存在することが非常に少ないさま。また、数少なくて珍しいさま (reference), so you can't use it with a negative as in the first example because it would become a double negative.