Don't concern yourself with old part numbers, as they change frequently without your approval. The vendor numbers stamped on parts often do not match the current manufacturing number at the time.
You have the correct belt, and the only one for all the Neptune series. Maytag found that the reason the old belts fail is that they stretch out, which is why your old belt is longer. To install the new belt, slip it first over the motor pulley and try to center it on the motor threads, then with the left hand hold the belt tightly onto the large spinner pulley at the top and turn the pulley counter-clockwise while holding the belt firmly on the pulley with both hands as you turn it.
You will not break the spinner pulley! Once the belt has snapped onto the spinner pulley, you need to center its track on the pulley by using the notches in the motor pulley to align the belt. Do this by turning the spinner with the left hand and moving the motor side of the belt back and forth on the multiple grooves until the belt is more or less centered on the large wheel. Yes, it is hard, but it does go on there.
Since these Neptunes have no transmission, the drive belt must do all the work, therefore they will stretch out in a few years. It is recommended that, after installing the new belt, you set the machine to a spin cycle and let it spin at high RPMs to get it settled in.
Part # 12001788
Kit, Belt & Isolator, AP4009041