Slate's Seth Stevenson sings the praises of boxer briefs, and I agree with him. And like him I experimented with boxers in high school and college, but boxer briefs are the only way to go for these reasons listed as advantages by Stevenson:
Support. The obvious, yet oft-unspoken flaw with traditional boxers is their lack of cuppage. They are useless for athletic events, and can even be a hindrance. (An acquaintance refers to the "tunnel" created by wearing boxers under soccer shorts. Via this tunnel, one's testicles can gain sudden and direct access to the world outside.) Boxer briefs hold your goods in place and out of sight.
Stability. Traditional boxers never sit still. They are forever riding up above the waistband of your pants, or slipping down below it. That loose fabric tends to twist, and bunch, and wedgify. Constant realignments are required. (This is especially true with the "bubble-butt" cut of boxer, which uses a spinnaker-like central back panel. The idea is to avoid having any seams line up with the butt-crack, but all that extra cloth just crawls up in there anyway, to disastrous effect.)
Containment. That simple slit of a fly on traditional boxers encourages a phenomenon I will term "flop-out." Some boxer shorts seek to rectify this with a button enclosure, but a button is the last thing you care to deal with when you urgently need to urinate. Boxer briefs use the much more effective and user-friendly Y-front.
Aesthetics. My unscientific polling suggests that ladies dig 'em. While it has all the comfort, support, and fit of a knit brief, the boxer brief's full-cut thigh lends it the modesty of a traditional boxer. And that thigh is functional, too—its snug, ribbed cuff serves to hold the garment in place. This prevents the boxer brief from riding up or (worse) burrowing into one's posterior cleavage. (The Calvin Klein boxer brief is particularly well-tailored, and is my personal choice. I own one pair of boxer briefs from 2(x)ist, bought at the little store in my gym when I forgot to bring a change of underwear, but I find they take an overly presentational approach to the genitalia. Sort of a push-up effect.)