幾 simply more literary/archaic than
幾 only be applied to certain words whereas
何 is more versatile?
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As far as the meaning, there is no difference. To say otherwise would be nitpicking.
The only (and important) difference is the way the words using 幾 and 何 as suffixes sound in our native-speaking ears. It is the classic "softer-sounding kun" vs. "sharper-sounding on" difference.
For instance, 幾重（いくえ） sounds much softer than 何重（なんじゅう） though these words mean exactly the same thing --- "multiple folds" or "how many folds". One would opt to use the former in a poem or story, but if one used it in a daily conversation, one would look like a weirdo.
One could say 何 is more versatile because that is what we use much more often in real life.
I think they have just slightly different nuances. They are very similar; but, I think that 幾 has a more "specific" nuance (as in with relatively small numbers), whereas 何 has a more "broad" nuance (as in "to what extent" or "why").
For 何度, デジタル大辞泉 says:
For 幾度, デジタル大辞泉 says:
The entry for 幾 says:
And the entry for 日中 of 幾 is interesting, too:
Maybe one could think of it as "how much" (何) vs. "how many" (幾)...