Tangentially, some car manufacturers have done prototypes of this as well. Look up the Porsche GT3R Hybrid or the Volvo S60 prototype for an example... Unfortunately it appears that for one reason or another no one has decided to move forward with these into actual consumer cars.

There are a few reasons including weight, safety, cost, package size, etc. A flywheel that can hold appreciable energy requires high speed or high mass as well as a low drag / bearing friction environment. The passengers must be protected from catastrophic failure of the flywheel by some strong casing ( more weight ) and cant be 'turned off' in an emergency. In race cars where heavy braking and acceleration are the norm and cost is not a factor, light, strong, fast flywheels can do a good job provided they run on active bearings in a low pressure casing. In a heavy road car the additional weight required makes the engine less efficient in terms of mpg, and probably less safe during a crash ( or if similarly safe, much heavier due to casing requirements ). Combine this with the inability to externally charge the flywheel and there arent really any great reasons to use it over electric energy storage