what are the advantages of controlling servos through a dedicated serial controller board (like the Pololu) against using the Arduino alone for that job?
One major advantage is that your Arduino will have much more processor power left over for other tasks.
frequently when using servos you are controlling something that needs a lot of calculations or processing of inputs, this can be difficult if the processor has to update the servos all the time.
If you're only running a couple of servos, there's very little advantage to the external board, unless you're really hurting for lack of I/O pins.
If you're controlling several, it saves pins and, as MikMo pointed out, CPU time (because the library for running many servos does a lot of work in software that's handled in hardware with the "basic" Servo library).