I like them both, and yet have very little desire to wear either colour at the moment. I think I overdid it on both counts. I went through an orange phase in my twenties (I had a memorable traffic-cone orange hoodie and an outerwear vest reminiscent of the gear for airport runway personnel) and I think I got it out of my system. And through most of my twenties and early thirties, I seemed to have a lot of burgundy -- burgundy pants, burgundy tops, and several burgundy pairs of Doctor Martens.
I guess I'm going to have to go with Team Burgundy, because I currently own no orange, but I do have burgundy rock n' roll Fluevog boots, a pair of burgundy Converse, and a pretty 1940s-style burgundy dress that I wear a lot in the fall and winter. I won't be adding any burgundy to my closet, though.
One thing about burgundy: I won't wear it in the spring or summer. It feels all wrong.