Well then, to start, there's the serenity of zooming along on a spring day with that Zen-like oneness... oh wait, that's for the nude cycling topic.
Er, it's different, see. There's only one gear. It doesn't go to eleven, or ten, just one. I find it a slightly different skill to be mastered, rather than anything life changing; I ride 2 or 3 bikes in a week, 2 fixed and one geared at the moment, and have no trouble adapting from one t'other, and have no particular preference, but fixed gear riding is a slightly different art. It's good to have a drivetrain which is clinically precise and efficient, and good to 'feel' the road and nuances (wind, slight gradients) through a drivetrain which keeps you moving forward. It seems to preserve momentum which is hard won - some call it a flywheel effect as your legs go over TDC [top dead centre] and it's probably good for your riding style in terms of cadence and smooth style, using a gear to learn of what you are capable instead of automatically downshifting for a small hill, and getting a good feel for the bike and how to handle it over bumps and potholes etc. The ability to moderate speed with the legs is also an advantage, using leg braking and 'soft pedalling', and slightly sore legs and learning how to get the best out handling the bike took a few months. Downsides, not that I can think of, provided you're not caught miles from home riding into a 25mph headwind. Singlespeed, which has its place, is in no way comparable. I rode SS (chain looped over the 15t on a 5 speed freewheel) for less than a week before I invested in a fixed sprocket.
Problems: when riding in traffic, it helps if you're good at clipping in, either side. I had to learn, especially in traffic, to keep clipped in as I slowed, and then unclip and put the foot down in one smooth motion, having the other in the 2 o'clock position ready for takeoff. I would not, under any circumstances, ride a fixed without a well connected pedal system.