I don't buy liner lock knives anymore, so for me this is an easy choice. For someone willing to buy a liner lock, it becomes a *really* hard choice, though. The EDI Genesis is an outstanding product in all respects, including the responsiveness of customer service all the way up to owner.
The Axis, at this time, is my favorite 4" blade folder in the world, bar none, including any and all custom folders. The combination of lock reliability, blade performance, and handle comfort even in prolonged use appears to be unmatched.
Both knives look great. Both knives have excellent ergonomics, although the finger cutout on the Genesis bothers me a little bit (a few other people have reported the same thing; most people aren't bothered by it at all). Both knives are made from quality materials.
Clip: I love that Genesis clip. Love it. The Axis's clip is good enough, but the deeper the better is the way I feel.
Lock: Well, you know my feelings here. Although there are exceptions, long experience has made me not a liner lock fan. The Axis lock appears to be stronger and more reliable. It's also ambidextrous.
Blade: Both blades are exceptionally good shapes, both for utility work and defense. The Axis's blade is recurved, and recurved blades provide better slicing performance than non-recurved blades, provided the edges are at the same thinness and grit. However, the Genesis' non-recurved blade will be easier to sharpen. This is a straight performance (Axis) vs sharpenability (Genesis) tradeoff.