what is wrong with B is it because of "was"

In D and E, you can tell determined = participle, not verb, because (a) it immediately follows a comma and (b) there's no subject you could match up with determined in a normal subject-verb configuration. Even if you match determined with the noun "times" (to which the clause is referring) "times, determined by" is not proper subject-verb config.

Why differed in D cannot act as a participle

Ron already explained this one

This is a toughie - it looks like you've realized that the key to the problem is parallelism, but you've been tricked by the rather weird form of parallelism exhibited here.

The only difference between choices D and E is differed/differing, and, as you've no doubt realized, the key issue is making the choice that's most parallel to 'determined' (from the first half of the modifier).

Here's the key: 'Determined' isn't a verb here. It's a PARTICIPLE: a verb form that functions as an adjective.
The local times were WEIRD (adjective).
The local times were DETERMINED (participle acting as adjective) by ...

So you need to pick the second half that's also 'participle acting as adjective'. Try them both:
D) The local times were DIFFERED: uh oh, that doesn't work.
E) The local times were DIFFERING from city to city. That works. It's a 'present participle' rather than a 'past participle', but it's the only one of the two that's a participle at all.

hope that helps.