tldr; use an appropriate-encoded string unless there is a reason to do otherwise.
It depends on what is required. Here are some differences - keep in mind that modern computers are fast and conversion is likely only a small fraction of overall application time so "more processing" is generally not even be applicably measurable!
String (with ISO 8601 or the stricter XML dateTime):
- "more space" / "more processing" (see above) / fixed size or variable size
- standardized culture-neutal format
- human readable and easily identifiable
- supports timezones
- more range (-9999 to 9999)
- more/arbitrary precision (up to 1us)
- lexicographically ordered (within same timezone and compatible format)
Epoch (UNIX variant):
- "less space" / "less processing" / fixed size
- standardized culture-neutral format
- not human readable (a diligent coder should be able to identify "about now")
- no timezones (can't even distinguish between "local" and UTC)
- less range (1970 to 2034 with a signed 32-bit number)
- less/fixed precision (1 second)
- numerically ordered
(The Julian day is another encoding with many similarities to an Epoch time.)
Conclusion:
Unless space/performance is a proven issue - this requires a performance analysis and functional requirements - I'd pick the former. Computers today are a good bit faster than computers just a few years ago and much, much faster than computers decades old.