What are the pros and cons of driving a bipolar stepper motor with a motor driver IC, like the TI DRV8824 (perhaps packaged on a carrier board like Pololu's http://www.pololu.com/product/2131), vs. using the Arduino Motor Shield based on the L298, or perhaps just an H-bridge as documented in the links in the documentation for the Stepper Library?
Likely that TI chip is more efficient than the L298 - so that would be one thing it would have going for it.
On the other hand - in theory using the L298 (or better, your own custom driver system using discrete components or some other mosfet-based h-bridge or half-bridge chip) - driving it directly with the Arduino would potentially allow you to experiment with your own software-based chopper driving system (via code and such - you would need to implement a current feedback monitoring system, too) - different stepper drive modes, etc. Again, in theory, all of this could be done going that route.
If I used a 8824 or similar driver chip, would I be able to use the the Stepper library, or the AccelStepper library?
Unlikely - as you have noted:
It looks like these libraries are designed to handle controlling the 2 stepper coils directly, while the 8824 would do this itself, providing higher-level direction and step inputs.
Yes - the 8824 does everything for you. I bet if you looked, you could even find someone who has already created a library for the Arduino to make interfacing with the Pololu 8824 fairly easy - if not, such a library wouldn't be difficult to write (and would help others).
The thing you are likely missing about the L298 is that in the case of stepper motors, it wasn't meant to be used alone except in really low-cost applications (such as a DC h-bridge controller or for stepper motor uses where cost was a main factor) - there is a companion chip called the L297 which is meant to be interfaced to the L298, and it provides a lot of additional functionality to the L298 when used for stepper motor applications (take a look at the L297 datasheet, and reference the L298 datasheet side-by-side to see what I mean).
Even so - the pair of chips likely doesn't give everything that the 8824 does, it will have a larger layout on a PCB, and it won't be as efficient.
Then again, if you used that pair of chips, you still couldn't use the existing Arduino stepper libraries with it, because they weren't designed for that combination.