Does a 2 stop filter have the same result on your picture as subtracting 2 stops with ISO/aperture/speed? What would be the purpose of the filter if you can decrease the light by other means with the advanced digital cameras of the day? Does it have any other features or uses?
Yes neutral density filters work the same with digital cameras as with film cameras...maybe better since digital cameras can readily be white balance adjusted for any "non-neutral" color cast in a filter (particularly cheap ones) if the photographer is interested in setting the camera's white balance.
Neutral Density filters allow using wider apertures with and/or slower shutter speeds to achieve a photographer's intent. Digital cameras are generally better at allowing wider apertures at higher ISO's (via faster shutter speed) and/or narrower apertures at faster shutter speeds (via higher ISO's).
Modern cameras are good at handling low light.  ND filters address too much light.
 One thing that often separates cinema cameras and lenses from stills cameras and lenses is built in neutral density filters to handle over-lit situations.
Does a 2 stop filter have the same result on your picture as subtracting 2 stops with ISO/aperture/speed?
It might have the same effect on the overall brightness of your image, but with different aperture settings or exposure duration settings you can't get the exact same picture with the following exceptions.
- If your subject is a flat, two-dimensional object that is perfectly parallel to your camera's image sensor, then minor changes in the aperture will not change the picture that much. Lenses do get at least slightly sharper or blurrier as the aperture is changed, though.
- If everything in the camera's field of view is perfectly stationary with no movement and your camera is also perfectly stationary. Lengthening or shortening exposure time won't have much of an effect upon the picture you'll get.
On the other hand, if anything in the frame is moving then changing the exposure time will alter how far the subject moves during the exposure. Or if you close down the aperture to allow less light into the camera, you increase depth of field and increase image degrading diffraction.
Changing the ISO setting does not increase or decrease light at all. It just changes how much the analog electrical signal created by the light that strikes the sensor is amplified before that analog signal is converted to digital information.
Glossary for the abbreviations that are used in what follows:
- Tv = Time value, exposure time, "Shutter Speed"
- Av = Aperture value, f-stop, f-number
- ISO = Analog signal amplification setting before conversion to digital information.
- "Base" ISO = The lowest analog signal amplification level the camera offers, usually ISO 100
What would be the purpose of the filter if you can decrease the light by other means with the advanced digital cameras of the day?
The purpose of ND filters is to reduce the amount of light captured by the sensor without changing Av or Tv when you are already at base ISO which can not be further reduced.
You might want to keep the Av low and/or the Tv long for specific reasons.
If you are shooting a waterfall and want a long exposure to show the blur of the water as it falls but you can't use 15 seconds Tv, even at f-22 and ISO 100 because the sunlight illuminating the scene is too bright.
If you are shooting a portrait outdoors and want a narrow depth of field to isolate your subject from the background, but even ISO 100 and 1/4000 Tv is too bright to use an Av wider than f/5.6 or f/8 with your 85mm f/1.4 lens.
If you are shooting a portrait outdoors and want a narrow depth of field to isolate your subject from the background while using flash, which usually limits you to your camera's X-sync exposure time, usually around 1/125-1/250 for most modern digital cameras. ISO 100 and 1/200 Tv is not going to allow an Av anywhere approaching the maximum for your wide aperture portrait lens. It may not even allow for proper exposure using the lens' narrowest aperture such as f/22 or f/32, which will also be fraught with problems due to diffraction.
For any given specific image result, there is only **one combination of settings that will generate that exact result. There are three factors which affect that result; and their primary functions are:
- Aperture controls image sharpness and relative sharpness (lens resolution and depth of field).
- Shutter speed control recorded blur/lack of blur (subject motion/camera shake).
- Light intensity/direction controls the resulting exposure of the previous two settings (and so much more).
One can think of aperture and ss as the technical aspects of an image (detail/resolution) and the lighting as the mood of an image (exposure, shadows, etc). Often, due to the available light, you cannot get the exact settings you would ideally want... so you have to make compromises. You could choose to compromise your SS or Ap (or both), or you can deal with the light.
Light intensity (sensitivity) is typically assigned as ISO; but that isn't really correct with digital. Increasing ISO just compensates for a lack of light; like a monitor brightness control does. Since ISO is compensating for sensor underexposure, it results in a lack of data and increased noise (it just makes the noise more visible/brighter; it doesn't cause it). Adding light would be much better if that is feasible.
ND filters compensate for excess light by blocking it; like sunglasses do. If an ND is too strong, and results in underexposure, that will also result in a loss of data and increased noise. In any case it is adding another optical element that was not part of the original lens design; which can (will IME) adversely affect resolution/sharpness/color to some degree.
**or within a reduced range of settings...e.g. you can have enough SS to freeze motion, and more SS will not make a difference in that aspect; but it will affect the other factors.