However, when observed with a telescope, the nebula appears green hued, and I can't see any of the characteristic red color of the picture. Where's the difference?
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High resolution astronomical pictures like this are usually taken in false color, in order to give them more scientific value besides being pretty pictures (I think they're more pretty this way, anyway :) ). Usually each color of the image represents gases of a certain kind, or a range of temperatures. Much of the details you see in these pictures aren't visible to our eyes, and instead glow in the infrared or UV/X-ray/Gamma ray. Usually infrared pictures appear silky and beautiful like this (warm gas in nebulas), while images of higher-energy radiation are normally grainy and mostly point-sources (GRBs, magnetars, etc).
In addition to false color images, when visible light pictures are taken of nebulas they are often very long exposures, or numerous shorter exposures stacked up together. Your eyes don't have this ability, so unfortunately you can only see small bits of detail in them.
Edit: Just a little extra info on this picture: The description says that this picture was constructed by hubble. It took 105 orbits to complete and used all of its imaging instruments. You can imagine why you can't get a view like this with a ground based telescope :P
At low levels of illumination, the human eye is incapable of detecting colour. It reports all the light as shades of grey-green. Only when the light reaches a certain intensity do the colour receptors react.
In 54 years of observing, including more than 600 deep sky objects, I've only seen colour in deep sky objects THREE times. In each case, this was with a large aperture telescope: 18-inch (Eta Carina Nebula), 22-inch (Dumbbell Nebula), and 74-inch (Cat's Eye Nebula).
Interesting point: If you were in space close to an object like the Orion Nebula, you would still see only shades of grey-green. That's because, even though it's much closer, it's also much larger, so the intensity per square millimetre on your retina is exactly the same. StarTrek's pretty coloured nebulae are wrong!
A final point: the colours in most photographs are real. The same colours were recorded back in the 1950s, when colour film was first used in the Palomar telescopes.
This is due mostly to the very small amount of light that is emitted by the nebula reaching us over such a massive distance. Our eyes are most sensitive to green light, so that's what we see first. If we could travel to the nebula, we'd see it as red too since we'd be getting more light from it the closer we got.
Cameras on the other hand can gather light for a long time to create a single exposure, so they pick up the full visible spectrum over long exposures.