What is the difference between absent and regrets at a meeting?

Answer 1

This is not hard and fast but in general:

Absent

denotes that no one knows why but the individual was not present, or that their absence was noted officially.

Regrets:

denotes the absentee notified the host and extended apologies for their inability to attend.

The latter acknowledges that the meeting and the time of the other participants attending is valuable and deserves that recognition, the former simply says "_____ was not at the meeting"