Neither.
Bird's eye is a type of figure that occurs within several kinds of wood, most notably in hard maple. It has a distinctive pattern that resembles tiny, swirling eyes disrupting the smooth lines of grain. It is somewhat reminiscent of a burl, but it is quite different: the small knots that make the burl are missing.
Flame is a description of a type of figure within the grain. It often refers to a figure with high levels of chatoyance which is that three dimensional shimmer some woods have. Much like the look in tigers eye quartz. It is how the light reflects off the grain at different angles. Some also might consider or describe the line between the sapwood and heartwood as flame. These terms can get confusing and it is always best to see the flame in person to fully appreciate its beauty.