Simply put, the internal resistance varies all over the place between the capacitor types. You can model it as a resistor in series with an ideal capacitor. When current runs through that resistor, it heats up. But the internal resistance is distributed "everywhere" inside the capacitor.
Higher internal resistance means higher power dissipation whenever the capacitor charges or discharges (i.e., current is flowing through it).
Heat kills the capacitor internally, either gradually, or suddenly/catastrophically.
The reason to use motor run or start capacitors is that they provide a second shifted-in-time phase for the motor, so that the windings can generate torque in two dimensions. One phase can only provide pulsating force in one dimension (let's call that X axis). Another phase, optimally 90 degrees apart (Y axis!) is needed to create the rotating vector (defined by sine and cosine). The capacitor creates this time-shifted second phase (not actually 90 degrees, but close enough for it to work) by storing and releasing energy. Within each storage and release event, there will be losses that heat up the capacitor.
Run cap provides this second phase all the time. In start cap systems, the second phase is only available shortly, because even the crappy 1-dimensional pulsating force can keep the motor running once it has inertia. Think about 1-cylinder engine.
Run capacitors charge and discharge at the mains frequency, i.e. 50-60 times a second, all the time, possibly for decades. Capacitors must have internal resistance low enough so they don't heat up internally.
Start capacitors can be more lossy to save on cost and size because they see a few seconds of use at a time, and possibly just some hundreds of cycles within their lifetime.
Always remember capacitors have current/power ratings as well because they are far from ideal.