Mounts are standard of 2 types: post mount and IS mount. Adapters are common from IS to post. Most frames and forks are IS brake mount.
Mechanical disc brakes are low braking power, and require more frequent maintenance because of cable stretching or breaking. Hydro brakes can be operated with one finger, even stopping a heavy bike at high speed, and the best of them require only yearly maintenance.
Pad surface is what matters the most, then lever modulation and adjustability. Large pad surface does make less heat, last longer, with better consistency, and improved modulation. Good levers have excellent modulation, giving the rider a precise control of friction up to a predictable locking point. They also have bite point and travel adjustment that is letting the rider tune the lever to exact same feel and performance as the pads are wearing.
Road or MTB, is mostly a matter of size and weight. Basically, they are the same within a brand, only bigger for MTB. On mechanical they have more leverage. On hydro, bigger pistons, more pistons, larger pad contact surface. Better cables or hoses are a good investment, at the first maintenance or pad replacement. Better pads too, especially for those riding dirt and/or high speed.