Difference between the M7 and M8 coprocessors?
As you noted, the M8 coprocessor is able to collect data from the new barometer in the iPhone 6 series phones as well as the iPad Air 2, which provides a measure of relative elevation for floors climbed.

We may also speculate that it is more accurate and power-efficient, although whether accuracy improvements are up to the sensors or the M8 is hard to prove (software improvements would also come into play here, as CoreMotion attaches a confidence value to measurements, which could be tweaked in software).

However, Phil Schiller raised another point during his presentation that many people seem to have missed: the M8 chip is now also able to recognize cycling activities. This new capability is reflected in the updated CoreMotion API, as detailed here:

https://developer.apple.com/library/ios/documentation/CoreMotion/Reference/CMMotionActivity_class/index.html#//apple_ref/occ/cl/CMMotionActivity

The M7 only recognized stationary, walking, running, automotive, and unknown. The M8 adds the cycling property to the list.

It is worth mentioning that iOS 8 does not make use of this functionality, and that the iPhone 6 does not write cycling data to the Health app on its own. You will need a dedicated app for that.