What would cause bands to appear lower on a nonreduced SDS-PAGE gel?
The major effect of changing the redox environment of a protein is the formation or breakage of disulfide bonds. Under sufficiently reducing conditions no disulfide bridges will be present, while under oxidizing conditions your protein will form disulfide bridges if it has the ability to do so.

Disulfide bridges can significantly change the tertiary structure of a protein and change the behaviour of the protein on a gel that way. Generally, disulfide bridges should make the protein more compact and make it run faster on a gel.

It is therefore plausible that your faster band is the protein with disulfide bridges, which will appear to run like a smaller protein as it is more compact. The slower band then is the protein without disulfide bridges which can adopt a more extended conformation.

I'm assuming you're talking about native gels here, as denaturing gels usually contain DTT or beta-mercaptoethanol in the sample buffer to reduce any disulfide bonds and avoid this effect altogether.