Physics dictates you can lift more with a barbell (load distribution, changes in muscle contributions, stability, etc. all play a role), so from a strength standpoint, you can't beat barbell bench. And I say it a lot, but once the neurological gains of a movement are pretty much tapped out, the only way to increase strength is to increase muscle cross sectional area (size more or less). Both can accomplish hypertrophy, but I prefer barbell at lower rep ranges (3s and 5s) and dumbbell at higher rep ranges (10s). Also, variation in what implement I'm using is important for staving off "monotonous overtraining"--i.e., your brain becoming tired of performing the same thing over and over and over and...
For future reference, dumbbells are also useful for when someone has shoulder issues. You're more able to adjust your elbows (more tucked, hammer grip, etc.) to take the pressure off the shoulder. If it gets to the point that dumbbells are hurting, though, it would be wise to see someone about the shoulder.