The cons are mostly, as has been mentioned already, related to the omega springs that make the lock work. They are thin pieces of metal, and they (rarely, but occasionally) break. Benchmade does not sell replacements, but they do offer a warranty, and some people have made replacements out of guitar springs.
Also, some people argue that the axis lock does not function well in sandy or gritty environments, since the mechanism can get clogged to some extent.
The pros are that, in addition to being literally ambidextrous--that is, there is a thumbstud on either side, the clip can be placed on either side, and the lock is accessible in the same way on either side--the AXIS lock is theoretically stronger than many other locks. For it to fail, the blade must shear through the metal pin that forms the lock, where with a liner lock, the blade and the lockbar only need to slip apart. For most users this never has any practical importance, but it can be nice to know your lock is one of the strongest invented.
Also, the AXIS lock is very smooth, both opening and closing, and it is safer than a liner lock, since you do not need to move your thumb across the path of the blade to close it.