'Advantage' is probably the wrong way to put it when it's more a case a of 'no longer being at a disadvantage.' LSN for the last two cycles (esp. compared to those of the boom years) shows that the numbers that get you in in Sept. are the same as the ones that get you in Jan/Feb. In the schools I was looking at, it appears that there's a little better shot at $ if you apply early - but that might also be some confirmation bias because that's what in theory I'd expect to see.
If you look at LSN trends this cycle, it appears that schools (T14s I should say) are waiting it out with their fringe, 25%-50% candidates as they get a better idea of their app pool. Regardless, it's a good time to apply to law school.