(I feel absurdly as if I had just wandered on-stage with an "another view" sticker on my forehead.)
I would take it as follows:
1. Worrying about measuring time precisely, and being paid for units of time, is what distinguishes the West from many other civilizations.
— here, I read "worrying" as applicable only to "measuring time precisely".
2. Worrying about measuring time precisely and being paid for units of time is what distinguishes the West from many other civilizations.
— here, I can read "worrying" either as in #1, or as applicable also to "being paid for units of time". However, the latter doesn't seem as likely to me, since "worrying about being paid" (= pragmatic perturbation) seems to belong in a slightly less elevated context than "worrying about precise measuring" (= science).
I find both a plural and a singular verb possible; the singular verb expresses a link between the two distinguishing things.
(I'll shuffle off stage right now, and let someone else have a go.)
MrP