Start caps are normally electrolytic types because these pack a high capacitance into a small can. They have very wide manufacturing tolerances and drift over time, so one cannot rely on a precise capacitance value and the motor design must allow for that. A start capacitor that is some way off the ideal value will cause reduced starting torque due to incorrect winding current and phase shift, but as the start winding is only in circuit for a few seconds at a time, it doesn't matter if the current is a bit on the high side, so long as the motor starts. Neither the start winding nor the electrolytic capacitor are continuously rated anyway.
Run caps are usually film types as these are continuously rated, and accurate and stable in capacitance. The maximum capacitance is much lower for a given size of can than with electrolytic types, but that's not usually a problem as the required value for run duty is much lower anyway. Accurate capacitance is important because it affects the winding current and phase shift all the time the motor is running. If the capacitance is too high, the phase shift will be too low, torque will be reduced, current will be too high and the motor may eventually overheat.
Just how critical the run cap will be, depends on many details of the motor design and use. On a well- cooled, generously sized motor running light you might get away with a much greater error in capacitance than with a low cost, hot-running motor that is barely big enough for the job.
Re. The fan, it's possible that the winding was already damaged by the original cap failure, and has shorted turns that are now overheating, rather than just on account of the 3uF cap.