Did/done

They are almost the same to me.
It seems the only difference is that the "apologize for what i've done" is more related to bad consequence at present time, however, "apologize for what i did" may have a bad consequence after "i did the thing".

[ W ]hat is the differen[ ce]  between these two? apologize for what [ I ] did and apologize for what [ I ]'ve done

In the first case, the speaker does not regard what was done as something that strongly affects the present situation.  He regards it as a past event whose immediate effects are mostly past.
In the second case, the speaker senses that what was done strongly affects the current state of affairs.  He regards what was done as a relatively recent event with important consequences for the present.

CJ