There's no direct advantage to a-speccing the races except you'll spend less money because the b-spec driver tends to be slow and therefore needs more mods on his cars. There are a few direct advantages to b-speccing the races: 1) you'll complete the game more quickly because you race at 3x normal time. 2) you can complete all the races without completing any of the license tests.
You will have to a-spec the missions though, and I believe that they require racing licenses to enter so you don't get away without doing the license tests at all.
There is an indirect disadvantage to b-speccing most of the races, and that is you get less experience so golding the licenses and completing the missions is going to be harder. If you don't do these you will miss out on a few prize cars.