If not, what is the difference between a screwdriver bit that is designed for a power tool versus the ones that come with a multi-bit screw driver set like these: http://www.homedepot.com/b/Tools-Hardware-Hand-Tools-Screwdrivers-Nut-Drivers-Screwdrivers-Multi-Bit-Screwdrivers/N-5yc1vZc98m ?
Best Answer
Screwdriver bits are generally considered consumable. Regardless of the brand, they eventually break/wear out. Hardware stores sell them in packs of 5-10 for exactly that reason. I wouldn't put too much stock in what kind of tool the bit originally came with.
The only exception is if you're using an impact driver you might want to be a little more picky. Impact drivers (or impact wrenches) put a huge amount of torque on the bit, and a cheap one could fail catastrophically. There are special "impact driver bit sets" you can buy that advertise high torque strength.