In short....it all depends.
We've all heard horror stories about someone who took their car to the dealership for service, and was given a seemingly unreasonable estimate for a seemingly simple repair. Maybe they tried to sell you flushes that you don't think you need. Maybe they quoted you $800 on a brake job that your neighborhood mechanic shop quoted you $150 for. What's the difference?
Well, there are several. Independent shops generally have less overhead. The money they make gets split up between paying for the parts, paying the employees and paying the bills. Independent shops have fewer employees to pay, smaller buildings that require less heat, water and power to operate, and usually a smaller advertising budget.
Dealerships, on the other hand, tend to have upwards of 20 employees, as well as a much bigger building to heat. Many dealerships are also run on a Fixed-operations system where the Service and Parts departments generate the money used to cover the Building maintenance costs, utilities, Fixed-op payroll and Parts transactions.
On the parts end of things, Independent shops generally source their parts through aftermarket vendors like AutoZone or Napa. These suppliers generally offer a budget line on many parts, including Brakes, filters, Chassis parts and such. So buying a set of front Duralast rotors and pads from Autozone might only set them back $95 rather than the $300 it would cost to get the Original Equipment level parts from a Dealership. In turn, while the shop might offer a warranty on labor, they have to work with the parts store to warranty out the defective parts. Usually not a big deal though.
Dealerships, however, prefer to sell their own brand, which should be kept in stock. Using the factory parts means that you are using parts that are designed exactly for the car, and in turn make the car work as engineered. Sure, you might be paying up to $150 per side on the rotors and up to $125 per axle on pads in parts alone, but You're getting a much better part. And, to be honest, if that OE part is going to fail, it usually happens within the 12 month / 12,000 mile warranty that is typical of an OE part.
Most cars require specialized tools for certain jobs. It might not make sense for an independent shop to buy those tools if they only see one such car come in every few months. Independent shops tend to work on everything. Some even specialize in certain types of cars. There are independent shops that specialize in European cars, some specialize in trucks or older carbureted vehicles, and others specialize in the more modern Hybrid-electric cars. Dealerships, on the other hand, tend to only work on the brand they sell. It is save to assume that a dealerships technicians are both ASE or state certified as well as certified by the manufacturer.
But where Dealerships really are better is for service while the car is under warranty. Not only do you get most things done at no charge to you (assuming it is covered), but you also get the benefit of knowing you are getting dealer-installed original equipment parts by dealer trained technicians. Most of the time.
But maybe I'm just biased because I work at a dealership that currently could use more work in the service department...or maybe not.