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.
