One of the biggest differences that you will see between the flaps on the two is that the NG (photo on the right) has continuous span flaps while the classic has split inboard and outboard flaps.
Combined with the longer wing on the NG, this gives the new flaps more surface area, so they don't need to be quite as effective. Overall, the NG actually has better performance than the classic, mainly because of the redesigned wing which is 25% larger and has winglets; both of which help out.
Even though the performance is better with comparable triple-slotted flaps, the double-slotted flaps are lighter (which helps with overall aircraft performance) and have fewer moving parts. This makes them more reliable and they require less maintenance. Most new aircraft designs are moving to either double or even single slotted flaps because of this simpler design and the fact that today's runways don't require the extra performance that would be gained by the more complicated flaps.
737 Classic Flaps