How to objectively compare the sound quality of two files?

If you truly want an objective measure, load both files into a tool that can do basic effects processing and mixing (Audacity for instance), and then invert one and mix them together. What ever is left is the difference between the two files - if they were identical, only pure silence should remain.

The most scientific way to identify changes or degradation between the two files is viewing the waveform on an oscilloscope.

You can use a audio editing program like Audacity to see if there is any clipping in either file. You may also want to apply a limiter or replay gain to the file.

Foobar has an ABX Comparator tool that will do a blind comparison of two audio files or clips. The listener votes a specified number of times as the tracks are played and the results are shown at the end so that the listener is not influenced by right or wrong answers throughout the test.

What you could do is to import both files in a audio editor such as Audacity. Invert one of the imported songs and you can now listen to the "difference" between them. Note that you have to be spot on e.g. the waveform need to be perfectly aligned for this to work.

From here on you can see what is "missing" (or added) to the source waveform and a spectrum analyzer or spectrogram will show you the details.