3-4 point, 5-4 approach, one-space low pincer, separate
When white separates with , Black and White seem to be a common continuation, at which point black must choose which side to develop, possibly with a or b.
The jump down to the second line is usually slow unless White has prospects for a severe attack, for example if White occupies the two adjacent corners as in the following diagram:
In a gobase search, this turns up as an old pattern, played by at least Ishida Yoshio and Maeda Nobuaki. Currently professionals do not like this result for Black, since he has invested lots of stones in what still not is an eye due to a, while White has a nice shape in the corner and sente.
If cuts here, then we get a similar pattern as above. Black can then continue by playing at a to capture one stone and stabilize his group at the expense of good influence for White. Another option for Black is b. The downside of b is that white's marked stones still has lots of aji so black isn't thick here. Sometimes white will save them immediately.
In fact, there are some other variations, for example can be extension at c to stop white taking the corner).
So, now lets compare these results to the (newer) joseki. In the new joseki white's corner is a bit bigger, but black is thicker on the outside due to the extension at . This extra thickness is generally judged to be of greater benefit to black than the extra points white got in the corner (afterall white is alive in both cases).
The reason is not the cut is then it would revert to the old joseki, which is not as good for black. takes a liberty off the black stone preparing the trade. With he abandons the corner in order to make the cut and get sente again to extend for thickness at , but before doing that black plays in sente (otherwise he extends at and white's eyes in the corner are gone).