The advantages are:
1. Very low maintenance. No adjusting cables, etc.
2. Excellent for cadence training. Because you cannot upshift, the only way to go faster is to pedal faster. A high speed, fluid pedal stroke is key to road and cross racing.
3. (Can be) Very light, easy to carry up stairs if you live on the 4th floor.
4. (Can be) Very cheap. A rear fixed wheel can be had for $100, and there is a ton of steel frame bike one Kijiji for about the same
Bonus - They are pussy filters. You can always tell who the bitches are by who pushes there bike up the Berri hill, and who rips up.
Disadvantages are
1. Hills. Both up and down. Unless you have pussy gearing (42x18) climbing is going to be a bitch. Once you get to the top, if you have a pussy gear, your legs are going to look like Sonic the Hedgehog coming down.
2. If you are new to them, you can get caught up forgetting you can't coast at a stop sign, usually causing a fall.
3. You will be stereotyped as a hipster.
Bonus. If you get in to track racing, good luck finding pants.