For instance, if I bring my eq down along the entire spectrum then adjust the volume up the same amount would'nt they just cancel each other out and nothing be changed?
EQ operates in the frequency domain. So, in theory, a digital EQ is continuously sampling your music and transforming it into the frequency domain where everything is simply a sum of sine waves. A parametric EQ is just a guassian curve centered on the frequency you choose with a width that you choose. By choosing the height of this curve and whether it is positive or negative, you define how much those frequencies along your curve are boosted or cut. You can think of a graphic EQ as just a bunch of individual parametric EQs spread along the frequency spectrum with predefined, nonadjustable frequencies and curve widths - you only get to adjust the amount of boost or cut. In actuality, it's more complicated that what I just said - the digital EQ's don't necessarily perform full fourier transforms back and forth between time and frequency domains, but instead they often use "digital filters" which are beyond my level of knowledge, but achieve essentially the same effect. Analog gear does it with a combination of capacitors,resistors,etc to make high-pass filters, low-pass filters, "tuned circuits", etc. And I'm already a bit over my head here.
But to answer your question, if you slide all the eq faders down but then boost the overall volume, you still have a changed sound since there is always granularity to that graphic eq - it's working at 10-20 discrete frequencies (so it will color the sound), whereas the volume control has a much more smooth and even effect across the frequency spectrum.