I know that road caliper brakes use shorter (and wider?) pads than v-brakes and MTB cantilevers, but I don't exactly understand the reason behind.
What would happen if I mount longer pads on road-style calipers and vice versa?
When you open the quick release on your caliper brakes, the brakes open wider to allow you to release your wheel quickly without deflating your tire. Longer brake pads could then touch the frame or fork and chip the paint. Some people even trim down their canti brake pads to avoid this scenario and sometimes in an attempt to reduce brake shudder.
Additionally, most v-brake style pads come with a series of washers (one concave and one convex that on either side of the brake arm you bolt to) that allow some angle adjustment which allows you to set the toe in and angle of approach on your brake pads without bending the brake arms. This is available on some brake pads intended for caliper brakes as well, but the washers tend to be much smaller. The extra washers may also prevent you from removing the wheel without deflating your tires.
But will these pads stop you? Yes