I still remember the cantilever brakes on my first early 90's rigid mountain bike, I loved the bike but the brakes ufff ! I did have a couple of hard times trying to adjust them properly. Next bike had V-brakes and so did many of the bikes I owned later on. V-brakes have enough power for most uses and the simplicity of maintenance is unbeatable. There is one flow though and I experienced it myself while touring in the Chilean Patagonia about 10 years ago. One rainy and muddy day, during a long down hill I couldn't manage to control my loaded touring bike. It ended up being a very scary and slow descent and my hands were sore from squeezing the brake levers so hard.
Even with their disadvantages, it wasn't easy for me to shift from rim brakes to disc brakes. I reckon because I was used to them and I knew exactly how to fully maintain them, they are also very inexpensive. But the power and control of disc brakes and specially hydraulic disc brakes is outstanding, and I could feel that from the first time I tried them.
Today I ride my Omnium cargo bike around Brussels on a daily basis. When the bike is heavy loaded and I'm rushing down a hill on a wet day I can only love the amazing power and control that a couple of hydraulic disc brakes can give. This is not a matter of fashion or marketing, there are good reasons why disc brakes have become so popular in the last years.
The top ten advantages of disc brakes:
Disc brakes have excellent stopping power thanks to their strong "bite"; big mechanical advantage and exceptional stiffness at the caliper.
Disc brakes perform well in wet conditions because they squeeze the water out very effectively with their strong "bite".
Disc brakes eliminates the risk of mud clogging because they are located at the hub, far from muddy tires.
Disc brakes don’t limit tire width because they are located at the hub. They don't bridge over the tires like rim brakes do.
Disc brakes are not affected by rim damage, wheel "out of true" or rim wear because they use a rotor as the braking surface.
Disc brakes allow switching wheels of different sizes, like 700C to 650B.
Disc brakes don’t need a braking surface on the rim. They allows for freedom of design, like eccentric rim sections or clean looking all-black rims.
Most hydraulic disc brakes have self adjusting pistons to compensate for pad and disc wear, which makes maintenance easier.
Hydraulic disc brakes have exceptional modulation and give excellent control.
Hydraulic disc brakes are low maintenance because they are sealed systems, oil change is normally less frequently needed than cable replacement.
What makes disc brakes so powerful ?
On disc brakes the caliper is much smaller than on rim brakes. The compactness and stiffness of a small caliper is more effective in transferring forces because it deforms much less than a rim caliper.
Also, a small caliper pressing onto a thin disc will have more mechanical advantage than a big caliper pressing onto a wheel's rim. Think of two pliers with same handle size, one has very short jaws like cutting pliers and the other has long jaws like a buffet pliers, which ones will press harder ?
For when it gets wet. Because of the great stiffness and mechanical advantage disc brake caliper are able to press so hard that they squeeze all the water out between the disc and the brake pads, so water doesn't decrease the grip.
For the specific case of hydraulic disc brakes the power is even greater because hydraulic systems multiply forces. For more details on this search for the Pascal's principle.
Any disadvantages ?
With disc brakes is easy to encounter compatibility issues because they use different types of brake pads depending on the brand and the model, there are dozens of different models of disc brake pads. Also, hydraulic systems use specific oil types and bleeding kits that aren't always compatible friendly. All those different standards make it difficult for bike shops to have a full stock of all brands and models, and for users to find the right spare parts.
Many disc brakes are more expensive than rim brakes. Generally speaking, a good quality pair of disc-brake calipers can be double the price of a good quality pair of rim brakes. Maintenance is usually more expensive on disc brakes, specially hydraulics.
Disc brakes, and specially hydraulic disc brakes, are definitely the best choice and the right choice for high performance braking like for mountain bikes, electric bikes and cargo bikes.
For most other uses the choice is less definite and it will be driven by factors other than pure performance and power. For a city bike for example price and easiness of self maintenance can be more important, so the choice could lean towards rim brakes.