Summary

In my opinion there is absolutely no point in using an ND filter with moon photography. The tutorial you mention uses the ND filter to reduce the glare on a human eye. By darkening the image, just as using sunglasses, you perceive a bit more of dynamic range then.

As you can just reduce the shutter speed in a camera, as you said, there is no use of this in photography.

Some scenarios for an ND filter

The idea with an ND filter is to be able to use a longer shutter speed. This can be useful in a number of scenarios:

• You want to use strobes in bright daylight and with open aperture. This will force you camera to use a shutter speed shorter than your flash sync speed. So to pull this off without resorting to other options like High Speed Sync (HSS), you can also bring down the ambient exposure via an ND filter.
• You want to create a dramatic feel with a clouded sky or moving water and create a dreamy atmosphere. With an ND filter you can prolong the exposure so that water or clouds move and become softer.
• You have things moving through your image at relatively high speed and don't want them to show up in you image. If you use an ND filter to prolong your exposure long enough, moving things might just be a ghostly haze or even not show up at all.

Why an ND filter is a bad idea here

What you usually never want is to prolong your exposure longer than needed if you shoot something that is moving. And if you just want to shoot the moon, possibly via a telephoto lens, prolonging that exposure will result in unsharp images, due to the movement of the moon within the frame - which is surprisingly fast.

So just to be able to get a sharp image with an ND filter, you would now need some gear to automatically follow the moon's movement like a star tracker.

What an ND filter does not do

Note that just using an ND filter does not magically enhance anything. The filter is applied to all levels of brightness the same way. So, there is no added quality whatsoever. It also does not change the dynamic range of you camera (how much detail is in the range from brightest to darkest parts). Quite the opposite: If the filter is not top grade, it might even add some color cast to the image, introducing reflections and flares etc.

Other types of ND filters

There are gradual ND filters that have a gradient in filter density, which an be used if you have a brighter part of the image and want to balance that with some darker part. I cannot imagine any use for them when photographing the moon alone, however it could be used to dim the moon to give more detail to the landscape the moon is over.