When your shutter is only open for 1/4000th of a second you will get a lot less light than when it's open for 1/60th of a second. Inside and outside.

The great thing about how cheap video is, is that you can experiment. Step outside and shoot cars going past. Do it once at 1/4000 and then again at 1/60. The other difference should be less blur on each frame when there is fast movment. Because the shutter is open for such a short time, you will see less motion blur. Or have a friend run past the camera - anything with movement. Then put that footage into your computer and play with the speed settings. That way you will know what looks better to you. And all it will cost you is a little time.

Let us know how it looks, okay?

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The aim of an argument or discussion should not be victory, but progress.
Joseph Joubert, essayist (1754-1824)

shutter speed can also be used as an effect. A high shutter speed will yield you a result like in Saving Private Ryan during the first battle scene where the camera seems to stutter. Of course, like the previous post said, there is less light coming into the camera so if you are shooting in doors, you'll need ALOT of light. iF you watch alot of movies with epic battles you'll notice this effect.

One thing to keep in mind if you are using a DVcamera with an adjustable shutter speed and iris is that you want to keep it on 1/60 unless you are going for a specific effect (24p is done in camera usually). If the lighting is too dark, try to fix it be adjusting the iris instead of the shutter.