Looking at it from a grammatical and not a measurement standpoint might help:
1) “some twenty feet away” is a count or actual measure of a distance, “feet” being used as a noun, while,
2) in “the twenty-foot drop”, “twenty-foot” (and, it should technically be hyphenatated because) the measurement is used as an adjective describing the noun “drop”. Technically, “100 yard sprint” should be “100-yard sprint” (though, hyphens are being forgotten more and more) AND you’re using the singular “yard” vs “yards” because the measure is being used as an adjective describing “sprint”.
So, it’s the same for distance and height/depth...all in how you’re using the measurement grammatically.
Why? Who knows? It’s just the set of rules I learned. There may be exceptions, as there are to almost every rule in (American?UK?) English, but, I can’t think of one right now.