The problem with ISO 8601 is that it isn't as simple as it looks. A lot of implementations that claim to support ISO 8601 do not parse all the different formats allowed by the standard. There are all kinds of weird edge cases like specifying a date with a year and week number that are valid.
RFC3339 defines a subset that is only the sane tinestamp format most people probably think of when they think ISO8601. It's a good alternative, although a lot of the time UNIX timestamps are honestly the most convenient option for everyone.
Source: spent a bunch of time designing a public API and thinking about this.