I'd suggest a paver driveway. Paver installations are flexible and last much longer than concrete or asphalt. You can even incorporate some of the newer permeable designs that let water through, and your redwood will appreciate it.
Here is the deal: no matter what kind of concrete or asphalt you put over those roots, it will not be flexible, and the tree is assuredly going to grow. If you install a segmented pavement of some sort, it will be relatively expensive, but down the line it will be cheaper because you can lift all the bricks in any problem area, re-grade around the roots as they grow, and then set the pavers back into position. Conversely, your new concrete is doomed to break eventually as the tree's roots continue to grow in diameter.
In some areas, you will also be getting taxed for each square foot of "impermeable surface" of roof, patio, shed, or driveway. A properly approved permeable paver installation will outlast your concrete, it will help keep the tree healthy, and might even pay for itself in stormwater runoff fees.
Installation costs are typically about 5 times (or more) higher than for concrete flatwork, though. If you do it yourself, you can have a fantastic looking driveway without much trouble. If money is a factor, just cut out the broken concrete and replace with pavers.
