A/C Fan won't turn on outside after replacing capacitor
Having a bad capacitor can put an extra strain on the motor.  An AC is actually a pretty simple device when it comes down to it, so the only replaceable parts (in this situation) are the fan and the capacitor.  Some units have a control board that handles some timing and sensors.  That can go bad but it's not as common (in my limited experience anyway).  If you hear the compressor coming on and the fan trying to start, then any control board is probably OK.

So possibilities:


The fan is in bad shape and somehow blew the new capacitor
The fan was damaged by the previous bad capacitor, ran for a bit, but now won't even start with a good capacitor
The new capacitor was just a dud and failed almost immediately


If you had a multimeter, you could check the capacitor and also check the resistance of the fan motor windings as additional troubleshooting steps.