Which would be theoretically a better encode?

[kikker]

If I choose a constant quality profile for an encode, the resultant file is 1 GB exactly. I then encode the same video entering a specific size (1 GB), but this time I use a 2-pass scheme.

All other parameters would of course be held constant. Which encode would have a better quality?

Also, what is the difference between the constant quality (--crf) and constant quant (--qp) modes?

[check]

if you are talking about x264, the answer is the 2pass. As for your other question, read the answer in the MeGUI FAQ (http://mewiki.project357.com/wiki/MeGUI:FAQ#What.27s_the_difference_between_Constant_Quantizer_and_Constant_Quality)

[PuzZLeR]

This debate between comparing the 1-pass option vs 2-pass, given the same bitrate/file size, and assuming ceteris paribus, is very detailed indeed.

Let me recommend you to a recent ongoing thread discussing this:
http://forum.doom9.org/showthread.php?t=125273