In my opinion, this is a silly debate that comes up over and over again. Both exercises are awesome, and both can/should be included in a good strength program. Of course, your goals will shape your program, but both dips and the bench are the best exercises for developing overall upper body strength.
I think the distaste for the bench press is overdone by many serious lifters these days because of how popular it is and how badly it is misused by the pretty boy teenager group in the globo gyms. Yes, the bench can screw up your shoulders if you overdo it without developing the rest of your upper body musculature. However, if one includes pull-ups and overhead pressing in their programming, the bench is a perfect complement to developing sheer brute strength in the chest, shoulders, and arms. I think the argument that it isn't functional is flawed as well. Yes, I know, there aren't many times in nature where I will be lying on my back pressing something over my chest, but I don't often snatch things over my head either and nobody is going to convince me to take snatches out of my routine with that argument either.
As to dips, I think their biggest advantage is the range of motion they allow and the shoulder flexibility that develops as a result of this range. Ring dips accommodate this more than bar dips, but both allow a range of motion that can not be accomplished with any barbell exercise. Dips, especially ring dips, help develop coordination and body control better than barbell pressing. While I don't think they develop the same brute strength as the bench press, the ability of the dip to develop relative strength is probably only equaled by the pull-up.
Quite simply, I wouldn't throw the baby out with the bathwater. The bench press is a great exercise that complements the dip and overhead pressing quite well. As balance seems to be a key part of elite fitness, rotating the exercises and developing all of them seems to be the best path unless you are a competitive powerlifter or gymnast.