Advantages: with no coasting, you can slow the bike down by putting backwards resistance on the pedals, therefore you don't need traditional brakes. It's also good for tricks, and it forces you to not be lazy on training rides, as you can't stop pedaling. Plus it's a bike in it's simplest form: two wheels and a crank, nothing else.
Some people get all sentimental about it and say "fixie riding brings you closer to the bike" or some stuff like that... Sheldon Brown was one of them. I'm sure you have to pay more attention, as you can't just daydream while you're coasting, you have to always give the fixie bike input.
I might build one for novelty, but it would not be a practical mode of daily transportation for me. You might build one and find out differently, who knows?