Advantages of Green
Green laser pointers have a shorter wavelength than red laser pointers: 532 nm vs. 650 nm. This allows viewers to see the beam mid-air, rather than just the red point landing on a surface. The development and marketing of the green laser enabled astronomers to use a laser pointer in the night sky, whereas red laser pointers could only be used where there was a solid surface, as in a planetarium. Green has the further advantage that the human eye is most sensitive to green light, due to the fact that we (and other animals) have more green-detecting cones in our retinas than any other color, as an adaptation to living on a predominantly green planet.
Advantages of Purple
Purple laser pointers are now on the market, joining green and red, with an even shorter wavelength at 405 nm. Because they're relatively new, purple laser pointers are sometimes perceived as cool, especially in geeky circles. Purple lasers also have applications outside astronomy, such as in DVD players and biomedical instrumentation.
Disadvantages of Green
Green laser pointers are more expensive and difficult to find than red laser pointers. While some people have managed to scavenge purple lasers from trashed Blu-Ray players, there is no cheapskate way to obtain a green laser pointer. Some are also concerned about the potential dangers of powerful green lasers used by untrained people
Disadvantages of Purple
Purple laser pointers are not all that useful in astronomy because the human eye is not very sensitive to the color purple. Against the night sky, a purple laser beam easily gets lost, and people have trouble following it. Also, purple laser pointers are much more expensive than green laser pointers.