I don't think anyone has mentioned this yet, and it might be out of the scope of your question, but one can ride brakeless on a fixed gear but one cannot (as safely) on a SS (although I know some crazies that do). Intuitively, you're going to burn a lot more calories and exercise a greater variety of muscles by letting your legs do most of the stopping, as opposed to letting friction do the stopping.

I've both a fixed gear conversion with a front brake and a brakeless track bike. The conversion has a brake because it has a nasty tendancy to throw the chain. Recently I've been using the brake increasingly often because I'm afraid that bunny hopping or skidding will throw the chain. Each bike has similar gearing. Anedcotally, I feel more "burn" after working out on my brakeless fixie than on my braked fixie.

If one is strictly comparing a brake-equipped fixie with a SS, I'd think that the difference in efficiency would be negligible (assuming that one exclusively used the brakes for stopping/slowing the fixie).