If I stand 10 feet away from a subject and I want a close up shot, I can either zoom in or I can physically move the camera as close as possible to the subject. Which one is a better practice, or is it the photographer's discretion?
Either can be the right answer, depending on circumstances. There is no one right or even usually right answer.
Moving in will change the relative perspective of objects. Things that are a little closer will appear disproportionately bigger, whereas from further back this size difference due to distance difference is reduced. A great example is getting close to someone's face, which causes the subject's nose to appear unusually large. Other times, that perspective difference is exactly what you want.
Zooming in has the advantage that it can be done from the same spot, and you don't actually have to get close to the subject. A obvious circumstance where this is important is when taking pictures of wild animals. They will run off or fly off, or in some cases try to eat you if you get too close.
Moving in allows a shorter focal length to be used, which allows for a slower shutter speed at the same level of blur due to camera wobble, like when hand holding. Moving too far in may not work with your lens. All lenses have minimum focus distances. If you really need to get up close, then you want a macro lens, extension tubes, closeup lenses, or a reverse mount.
Everything is a tradeoff.
Zooming changes the angle of view while keeping the spatial relationship between the camera and the objects in the image the same.
Moving changes the spatial relationship between camera and objects, but the angle of view remains the same.
Imagine that you're photographing a person that's 10' from the camera, and there's a tree 10' beyond the person, or 20' from the camera. No matter how much you zoom in or out, the tree is always twice as far from the camera as the person is. But if you move 5' closer, the person is now 5' away while the tree is 15' away; that is, the tree is now three times farther than the person. Or, if you move 10' back from your starting position, the tree will be only 1.5x farther than the person.
How does the difference between zooming and moving affect your photographs? Here are two important aspects to keep in mind:
Composition. If you want to minimize the distance between objects in your photograph, use a longer focal length from a greater distance -- move back and zoom in. If you want to emphasize the distance between objects, use a shorter focal length from a smaller distance -- move in and zoom out.
Depth of field. The relative distance to different objects in the scene obviously has implications on depth of field. As we know, smaller apertures provide greater depth of field. So, if for example you want the tree in the example above to be out of focus, you can choose a larger aperture (lower f-number), or you can keep the same aperture but move closer to the person. Conversely, if you want to put the tree in focus, you can use a smaller aperture (higher f-number), or keep the same aperture but back away from the person.
Update: I just spotted a post on Canon's blog about the advantages of a telephoto lens for portraiture, and it contains two good examples where the focal length changed and the camera was repositioned to keep the subject the same size. You can see a significant difference in the way the background looks due to the difference in relative distance.
The depth of field in typical "pictorial" settings is dependent only on the f-ratio and the magnification (object physical size / image physical size). When doing close-focus macro work this is not the case any more. So really only the perspective changes, if you stay with the same f-stop, shutter speed, ISO, detector. The artistic effects can be quite different, of course.
I now see AJ Henderson's point in the comments. The argument I made in this posting about the exposure is wrong. While with larger focal length (F) the field of view decreases approximately proportional to 1/F^2, at constant F-number N, the aperture is F/N, which means that the total amount of light per unit time on the sensor is proportional to 1/F^2 * (F/N)^2 = 1/N^2 which does not depend on the F-number.
This means that you have less room to decrease ISO etc. as I sugggested in the original posting, but you still have the advantage of less motion blur and depth of field etc.
Usually moving closer is better (as long you can still focus). By zooming less, you can shoot with longer exposure time without motion blurring due to camera shake. The larger field of view means that you are gathering more light, so this reduces the exposure time you actually need. This then gives you a lot of room to lower the ISO in order to reduce noise. With less noise, a lower noise reduction level is needed. Since noise reduction comes at the expense of resolution, this will also improve the image quality.
Response to comments:
If you zoom more, the field of view as determined by the image on the sensor becomes smaller (sensor size divided by focal length determines the field of view). So, the sensor will get less light. The reduction in the amount of light is approximately quadratic in the focal length. If I reduce the distance to the object by half, then I can approximately reduce the focal length by half so that it still fills the same part of the image. But then I'm going to have 4 times as much light. As far as focussing is concerned, without reducing the F-number, the hyperfocal distance will be approximately reduced by a factor of 4 while I have moved only a factor of 2 closer, so this is a net gain.
So, I would say that from a purely technical point of view where you focus on getting as much information from the subject in your image as clearly as possible, it is usually better to get closer to the subject. From an artistic/esthetic point of view you may well decide that such a picture looks ugly and a picture taken from farther away looks better.