Something that causes regret is regrettable.1 A person who feels regret is regretful, as is something expressing regret (e.g., a regretful letter of apology) or anything else that is full of regret (e.g., a facial expression or a morning after).2 These distinctions extend to the words’ corresponding adverbs, regrettably and regretfully.
Examples
Regretful
But even the critics declared that they liked Ike before they pronounced him, in regretful tones, the wrong man for the presidency. [Wall Street Journal]
Regretful of her own decision to not go see the Berlin Wall fall in 1989, Ghodsee wrote that she wants her students to witness history. [Bowdoin Orient]
Regrettable
Her claim that logging was a good thing for koalas was a regrettable start. [Sydney Morning Herald]
It is regrettable, however, that he will not be able to stand trial … [Globe and Mail]
References
1. http://www.oed.com/view/Entry/161396 ^
2. http://www.oed.com/view/Entry/161393 ^