From your own links to Brown-Driver-Briggs and looking down the lists of occurrences in scripture, there does seem to be a comparative difference in the two words.
Power (koach) appears to be vigour in procreation, vigour in the delivery of children, vigour in battle, vigour of land producing crops and vigour of various animals.
It seems to indicate organic power, power associated with flesh.
Context would indicate whether, or not, it is applied to angels or to God. Here, the context would not indicate that meaning. So, the above meaning, of organic vigour, appears to stand as relating only to human flesh on earth in all its vigours.
Might (chayil) is a matter of the display of strentth, of moral worth, of substantial wealth, of being numerous (as in an army), of royal power and of military significance.
It seems to be a matter of demonstrative strength, a display, an enforcement.
In Zechariah 4:6 both of these characteristics are contrasted with the way God works, in spirit. Not of fleshly vigour, not of organic life and not of outward demonstrations of physical might.
Note 1 : The single reference of roach meaning a kind of animal reminds me of the reference to leviathan in Job. This solitary expression of an animal with fortified scales, impenetrable skin and blastings from its nose seems, to me, to express a creature on earth (a crocodile, perhaps ?) but also to infer a spiritual, invisible entity of like characteristics.
It may be that we are to see roach in the same way.
Note 2 : Green's Literal (1994) is the Hebrew Interlinear I use personally. It is based on the Textus Receptus.