Do races advance your fitness as much as workouts?

Both.

Fitness isn't a single variable, its made up of many variables and different types of running affect these variables. Just like different workouts will affect you in different way, racing will affect you in a certain way.

Interesting thought. My uneducated, but rational response would be that the benefit of racing diminishes as the length of the race increases. The benefit of racing a mile outweighs it's taxing effects. However, racing a marathon has a much greater impact on the body than the benefits it provides. So, depending on the person, and obviously their ability to recover, the point where you're doing more harm than good falls somewhere in the middle. Maybe 10k for some, half for others. Some shorter, others longer. I would say, generally speaking, middle distances, provide greater benefit than breakdown. But frequency of racing makes an obvious difference, and like previously mentioned, fitness and base play a significant role.

George Young once said that there is no better preparation for racing than racing and I think he's absolutely right. But that doesn't mean you can't overdo it and wreck yourself. Ron Clarke raced pretty much weekly and in "season" increased that frequency. But Clarke did almost no anaerobic training which allowed him to rebuild his fitness between races and the races sharpened him.