There really is nothing magic about the bracket itself, but it is the entire philosophy that has been so revolutionary in orthodontics.
Here is what I have seen treating traditional cases and Damon cases side by side (excuse the technical mumbo jumbo):
The Damon system allows for there to be a sturdy tube in which the arch wires can glide around and work within. With the initial wires in the Damon System, all of the activity is on the outside of the tube and not the base, which is completely counter to how traditional braces work. In traditional braces the arch wires are tied into the base of the bracket to control teeth. Working along the outside of the tubes with light forces allow crowding to unravel much quicker by uprighting and de-rotating teeth simultaneously. This is more mechanically efficient than traditional braces.
Because the initial phases of treatment are targeted at uprighting and de-rotating teeth, the Damon System also produces broader more beautiful smiles that enhance your facial appearance. You can do this with other treatment systems, but a broad beautiful smile is far easier to deliver when the groundwork for it is laid down early in treatment.
The presence of a solid door holding the wires allows for rubberbands (elastics) to be started earlier with less adverse movements. This absolutely speeds treatment, especially when the space gained from uprighting teeth (as I pointed out before) is used for moving teeth back to front (or vice versa) at the same time with elastics.
Appointments are much shorter because there is no complicated means of changing archwires, and the little doors on Damon brackets are designed far superiorly in my opinion to other self-ligating brackets so there are less problems with the brackets themselves at appointments (and they should be since the brackets aren't cheap).
I don't need to drag patients out of school or parents out of work as often because of how long the wires can continue to work (Ormco has a patent on their special blend of wires). I respect my patients' time so I love being able to do this for them.
Customized torques in the brackets save time and appointments at the end of treatment when finishing up a beautiful smile to my standards. Damon Braces aren't the only ones that do this, but taken with everything else it is a much bigger deal.
It is important to understand that there is a substantial learning curve to applying the Damon Braces System, so it is wise to seek out an experienced Damon orthodontist to experience all the benefits of this philosophy. The good news is that Damon Braces have a great Doctor locator on their website that will make your search that much easier.