There's nothing magic about ultracaps, they're just one more technology that may or may not be useful for EVs.
Their main advantage is that they can charge and discharge extremely fast. So for a quick burst of acceleration, they're hard to beat. I suppose you could also regen much faster than with a battery pack, but I think generally a battery pack is able to handle any regen currents you want on and EV.
Their downsides are fairly important, too, though. First, their energy density (last I looked) was about 1/10th that of a li-ion battery. So they're really not very good for primary energy storage. They are making gains, though, and last time I looked was a while ago, so they may be much closer now.
Second, their charge/discharge curve is perfectly linear. That may sound good, but mostly it isn't. They go from maximum voltage at full charge to 0 volts when discharged. That's not very vehicle friendly -- you have lots of voltage to work with at full charge, but less and less as it discharges. A battery, conversely, is very non-linear...which again, is pretty much a good thing. For instance, the battery in my 2014 Zero SR (and many Zeros) goes from 116VDC at 100% SoC to maybe 92-95VDC at 0% SoC, with a lot of the curve lying between 100VDC and 110VDC. That's a lot easier to design a drivetrain around.
They might make very good supplementary storage, especially for a racing vehicle. They could supply a few seconds of ultra-high current when coming out of corners, which would be excellent. (In racing it's partly about going fast, but it's more about NOT going SLOW, so the harder you can accelerate out of corners, the better off you are.) But as Brian points out, it would add a ton of complexity, a fair amount of weight and a management headache for the electronics. Plus the batteries are getting better and better at supplying high currents for a short burst, so there may not be all that much advantage anyhow.
Ultracaps are fascinating, and wonderful for many uses, but just not really suited all that well for EVs, in my opinion. At least not yet.