To me, a sober alcoholic, one who's active in his or her recovery, implies some level of contentment in the person who is sober. Happy, maybe, certainly not miserable. Someone learning to cope with life without resorting to alcohol.
Dry drunk has a negative connotation, describing a person who is abstaining from alcohol, but who's alcoholism goes untreated. Someone who doesn't drink, but fails to cope with life in a truly healthy manner.