For passing times in JSON to/from a web API, why would I choose to use an ISO8601 string instead of simply the UTC epoch value? For example, both of these are the same:
Epoch = 1511324473
iso8601 = 2017-11-22T04:21:13Z
The epoch value is obviously shorter in length, which is always good for mobile data usage, and it's pretty simple to convert between epoch values and the language's local Date type variable.
I'm just not seeing the benefit to using an ISO string value.
Both are unambiguous and easy to parse in programs. The benefit of epoch like you have mentioned is that it is smaller and will be faster to process in your program. The downside is it means nothing to humans.
iso8901 dates are easy to read on their own and don't require the user to translate a number in to a recognizable date. The size increase in iso8601 is unnoticeable when compared to much much larger things like images.
Personally I would pick ease of reading over speed for an API as it will cut down on debugging time while inspecting values sent and received. In another situation such as passing times around internally you may wish to choose the speed of an integer over text so it depends which you think will be more useful.