To elaborate a bit on deltheil's answer:
- Signal to noise. For many applications of image processing, color information doesn't help us identify important edges or other features. There are exceptions. If there is an edge (a step change in pixel value) in hue that is hard to detect in a grayscale image, or if we need to identify objects of known hue (orange fruit in front of green leaves), then color information could be useful. If we don't need color, then we can consider it noise. At first it's a bit counterintuitive to "think" in grayscale, but you get used to it.
- Complexity of the code. If you want to find edges based on luminance AND chrominance, you've got more work ahead of you. That additional work (and additional debugging, additional pain in supporting the software, etc.) is hard to justify if the additional color information isn't helpful for applications of interest.
- For learning image processing, it's better to understand grayscale processing first and understand how it applies to multichannel processing rather than starting with full color imaging and missing all the important insights that can (and should) be learned from single channel processing.
- Difficulty of visualization. In grayscale images, the watershed algorithm is fairly easy to conceptualize because we can think of the two spatial dimensions and one brightness dimension as a 3D image with hills, valleys, catchment basins, ridges, etc. "Peak brightness" is just a mountain peak in our 3D visualization of the grayscale image. There are a number of algorithms for which an intuitive "physical" interpretation helps us think through a problem. In RGB, HSI, Lab, and other color spaces this sort of visualization is much harder since there are additional dimensions that the standard human brain can't visualize easily. Sure, we can think of "peak redness," but what does that mountain peak look like in an (x,y,h,s,i) space? Ouch. One workaround is to think of each color variable as an intensity image, but that leads us right back to grayscale image processing.
- Color is complex. Humans perceive color and identify color with deceptive ease. If you get into the business of attempting to distinguish colors from one another, then you'll either want to (a) follow tradition and control the lighting, camera color calibration, and other factors to ensure the best results, or (b) settle down for a career-long journey into a topic that gets deeper the more you look at it, or (c) wish you could be back working with grayscale because at least then the problems seem solvable.
- Speed. With modern computers, and with parallel programming, it's possible to perform simple pixel-by-pixel processing of a megapixel image in milliseconds. Facial recognition, OCR, content-aware resizing, mean shift segmentation, and other tasks can take much longer than that. Whatever processing time is required to manipulate the image or squeeze some useful data from it, most customers/users want it to go faster. If we make the hand-wavy assumption that processing a three-channel color image takes three times as long as processing a grayscale image--or maybe four times as long, since we may create a separate luminance channel--then that's not a big deal if we're processing video images on the fly and each frame can be processed in less than 1/30th or 1/25th of a second. But if we're analyzing thousands of images from a database, it's great if we can save ourselves processing time by resizing images, analyzing only portions of images, and/or eliminating color channels we don't need. Cutting processing time by a factor of three to four can mean the difference between running an 8-hour overnight test that ends before you get back to work, and having your computer's processors pegged for 24 hours straight.
Of all these, I'll emphasize the first two: make the image simpler, and reduce the amount of code you have to write.