All else being equal, processing power would effect the encoding time, but not necessarily the quality. Not that all else is necessarily equal, but one device may be able to process "x" number of frames per second, while a device with twice the processing power would just do the same thing at twice the speed.
Someone might be able to explain it better than I can (maybe someone who actually understands it
), but your example suggests compressing "raw" or original frames and outputting a compressed version, then re-encoding the compressed copy. If those compressed frames are re-encoded, the encoder doesn't know where they came from. They're decompressed, fed to the encoder as "raw" frames, and this time the encoder might also need to re-compress artefacts introduced in the first encoding process which it didn't need to compress the first time. The encoder might be able to encode the original video more efficiently because it's a little "cleaner" and easier to compress.
Whenever you use a lossy encoder to re-encode, there'll be some sort of generation loss, even if the exact same encoder settings and bitrate were used the second time.
http://en.wikipedia.org/wiki/Lossy_c...formation_loss
http://en.wikipedia.org/wiki/Generat...eneration_loss
As an experiment, if you have an image editing program such as Irfanview which allows you to set the quality when saving an image as a jpeg (which is lossy), try saving an image at 100% quality. Now open the second image and resave it as a new image at 100% quality. Then open image number three and save it as a forth image.... 100% quality again. You'll probably find with each successive save, the file size increases a bit.
Or you can save the image at 50% quality and reduce the file size substantially, open the saved version, resave it at 100% quality and find the 100% quality version is much larger again. No information was "restored" as a result though.