The difference is small because map is trivial, but the implications of using higher order functions isn't. Even something like trees sees a divergence between the styles (not necessarily in the order of assembly language vs. high-level, but certainly more than you're making out).

Higher order functions are really useful for splitting apart navigation of a data structure and the processing of the data contained within it.

Given that you can use purely higher order functions for control flow, I wouldn't say one is more essential than the other.