Both live in the same control box and cause similar symptoms — but they fail differently and need different tests. Here's how to tell them apart.
The start capacitor gives the motor an extra jolt of torque to get spinning from a dead stop. The relay is a switch that disconnects the start capacitor and start winding once the motor reaches about 75% of running speed — leaving only the run winding energized. A capacitor problem usually means the motor won't start at all. A relay problem usually means the motor starts fine but then something goes wrong shortly after.
Testing order: Always test the capacitor first — it's the more common failure point and the cheaper part. Discharge it, then use a multimeter's capacitance setting to compare the reading against the microfarad (µF) rating printed on the case.
A relay that sticks and fails to disconnect the start capacitor keeps voltage flowing through it continuously, which cooks the capacitor within days or weeks. This is why control box manufacturers sell relay-and-capacitor kits matched to your horsepower rather than individual parts — if one failed from age, the other is usually close behind.
If your control box is more than 8-10 years old and one component has already failed, replacing the entire box is usually the better call. The labor to open the box, test, and replace one part is nearly identical to replacing the whole unit, and you eliminate the risk of a second failure a few months later.