Fixed wheel machines tend to be significantly lighter than geared ones, and the fact that the chain will not unship when riding over very rough ground can be useful.
Cyclo-cross riders for many years often chose fixed wheel becuase you spend one third to half your time running with the machine on your shoulder.
After you have finished your cyclo-cross event a fixed wheel machine is very much easier to clean and maintain.
Without gear and all the parapahalia, fixed wheel machines can be very cheap to buy and maintain, and they are not as attractive to thieves who generally fall off in their panic to rush away.
Fixed wheel machines can be useful if the grip is uncertain, such as in frosty conditions, because applying the brakes can lock up a wheel readily in such weather, but on fixed you can slow down smoothly and very subtly.
Fixed wheel is usually excellent for specialised hill climb events - in these events you start on the lower slopes or the very bottom of a steep hill, and you are timed to a finishing line somehwere near the top - advantage is very light weight, you do not risk unshipping the chain when trying to change gear whilst maintaining drive through the pedals, and fixed wheel rolls around better than gears, it helps reduce the top-dead-centre effect.
You would be surprised at just how fast you can pedal when you know how, I have been able to spin fixed wheel downhill at well over 180 rpm.
You have a far better ‘feel’ for the road, the machine gives you better information about the road surface.
When you arrive at traffic lights, a good fixed wheel rider can often do a standstill trackstand, so you don’t have to put your feet down - its just a matter of pride than anything but it does look quite cool - you can do this on a geared freewheel machine but it is very very much harder.
You tend to ride fixed wheel with a ratio that is fairly low for riding on flat ground, so something like 48 tooth chainring to 18 or 20 sprocket is fine.
When we do time trial racing in the early season, we will often use fixed wheels for the limited gear events - these are event set up mostly for competition training purposes, and the size of the largest gear you are allowed to use is limited - this helps reduce the chance of early season injuries. On a geared machine its something of a pain to adjust your deraillers to ensure you cannot select too high a gear, but on fixed its easy, select your two cogs and off you go.