There are lots of differences between ISO 8601 and RFC 3339. Here is some examples to give you an idea:
2020-12-09T16:09:53+00:00 is a date time value that is compliant both both standards.
2020-12-09 16:09:53+00:00 uses a space to separate the date and time. This is allowed by RFC 3339 but not allowed by ISO 8601.
2020-12-09T16:09:53-00:00 has a negative sign in the time offset. This is allowed by RFC 3339 but not allowed by ISO 8601.
20201209T160953Z omits the hyphens. This is allowed by ISO 8601 but not allowed by RFC 3339.
ISO 8601 allows for things like ordinal dates such as 2020-344 which represents the 344th day of year 2020. RFC 3339 doesn't allow for that.
For your questions:
Is one just an extension?
No. As shown above each standard supports syntax variations not supported by the the other standard. So one syntax is not a superset or an extension of the other.
Should I use one over the other?
Of course this depends on your scenario. A safe general strategy is to generate date time strings that are valid by both standards.
Another good general strategy is to use an existing standard library for parsing/formatting date time strings and not write custom implementations unless you are addressing a genuinely custom scenario.
Do I really need to care that bad?
Well, that's up to you. Most regular developers who deal with date time strings should have a high level understanding but don't need to dive into the details.