Flywheels are useful when you have a fixed installation with no weight constraints because you get certain benefits over LiPo cells, namely longevity. As a matter of energy density though, they are awful compared to LiPo and could never efficiently be used in a vehicle. Also, the gyroscopic forces would be such that it would be impossible to steer the vehicle.
> the gyroscopic forces would be such that it would be impossible to steer the vehicle.
That's a solvable problem, and has in fact been solved by F1 teams that have used it - Mclaren won two titles with their KERS equipped cars. The cars are demonstrably far from 'impossible' to steer...
All you have to is mount two of them rotating in opposite directions to cancel it out. The axles need to be able to handle the load but the rest of the car sees nothing.
I could see it being used as a "quick-charge" portion of the capacity. i.e. you have a full-sized battery, but also a small flywheel so that if you're really in a pinch you can rapidly charge enough to get somewhere without waiting for an hour. It'd be like having a small SSD paired with a large HDD.
Angular momentum increases linearly with mass but with the square of the radius. So another key solution to capacity is to increase the size of the flywheel. Obviously there are real limits to this in a car.
I think it could be useful in congested cities with lots of stop and go driving. If there were magnetic flywheels in the street that mated with magnetic flywheels in the vehicle, energy could be transferred at stop lights. Provided the field didn't move the whole car, and if gyroscopic forces were mitigated, as another poster pointed out
I think the first year F1 had kers some of the teams used flywheels, but I think since then everyone seems to have moved to standard battery pack type systems, so I assume there are downsides to them.
At least if you drive a bit above the speed limit when commuting :)