Sounds like a bad run capacitor on the fan. Usually, there is a dual capacitor that is shared between the compressor and the fan. I had one go out about 2 years ago and did the same thing. In my case, both the fan and compressor would not start. You would hear a click/buzz at the outdoor unit whenever it tried to start the fan/compressor. I could also flip the fan with a stick, and the fan would run for a few minutes, but still no cooling as the compressor wasn't starting. I did a little trouble shooting and narrowed it down to the cap. Picked up a new cap for under $20, I installed it, and it was fine after that. Actually, I just installed a new system and still have this practically new cap. If it is the same type that your system needs, I would be happy to send it to you. I have no need for it.
In your case, you might either have separate capacitors for the fan and compressor, or it might be shared and only one side is dead. The other possibility is that it is a dual capacitor (that is totally dead), but your outside unit might have a hard start kit on it (which is basically just an extra capacitor that goes to only the compressor) and that hard start kit is enough to run the compressor. It sounds like the compressor is running, but the fan is not.
Another possibility would be a bad winding or similar in the fan motor.
Anyway, my guess would be run capacitor or condensor fan motor. It is hard to tell exactly from here. Neither one is very expensive, but labor/trip charge will probably be the largest part of the bill. The labor (in terms of hours) should be low for either of these parts. If you are comfortable doing some testing on your own, I can help guide you. Make sure that you pull the disconnect before removing the panel though!