Ball bearing:
Very similar to the Axis, although fairly new, has proven itself. It got off to a rocky start, and if you search the forums for ball bearing lock threads, you'll find a lot of them with people that have had problems with it. As far as I know, all problems have been solved. They are very dependable, and I have had no problems at all with it.
The springs in the BB Lock (mine anyway) seems a lot less stiff than that of the Axis lock, this means there's much less resistance when opening/closing. It also seems a tad more smooth than the Axis. Since you can see the tang, you can just put a drop of oil from the side, and work it into the action.
I think the ball is a little too small. The blade stops on the backspacer, instead of a backbar (like the axis), so it makes cleaning a bit more difficult.
The Axis.
Well, what can I say. It must be the favorite lock of the knife world. Very dependable, solid lock-ups. They shrug off lint and crud that gets on the tang, easy to clean, easy to lube. The springs on the Axis bar, may feel a little stiff in the first week or so, play with it a lot and it'll soften up a bit.
Both are ambi, and work well both ways.