(snip)
1) If you got the white balance wrong, with a RAW file you can correct it without losing as much information in your picture.
2) You can selectively sharpen parts of your image rather than the entire thing.
3) When photographing very saturated colors (like flowers), it is easy to exceed the color range available in your image and lose detail. By controlling the saturation yourself, you can often avoid this problem.
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Photo workflow packages like Adobe Lightroom and Apple's Aperture have made dealing with RAW files very easy. In Lightroom treats RAW and JPG files the same way (not sure about Aperture), so they are no more difficult to work with.
white balance is one of the best reasons to shoot raw. even though all modern cameras have 'auto white balance', they are generally 'templated' against sunlight, incandescent, flourescent, overcast, etc. the problem is these light sources are all different temperature values with varying amounts of green to magenta. i adjust the white balance for all of my shooting conditions and batch apply this to the appropriate images during a shoot.
the second is the abilility to avoid using the camera's sharpening routines (to make up for the anti-moire filter) and using a better sharpen technique such as Unsharp Mask or similar. as Mark points out, it can be selectively applied as desired
the 3rd point Mark raises regarding the application of saturation is also very important - as is application of contrast. by applying these to a .jpg in camera, detail can be lost and colour tone can be shifted, and this is not something that can easily be corrected in post processing.
one item that has not been raised is the lossy nature of .jpg files. averaging and smoothing are natural especially if the filters used to convert to jpg are not edge protecting or a decent algorithm. that detail can never be recovered and if the files are saved again, further loss is introduced. a '10' quality .jpg setting is often as large as a raw file (because most raw files have lossless compression built in) and yet they do not have all of the colour information.
ah, and yes... Aperture is like lightroom in that it treats all supported file formats the same - jpg, raw, psd, tif, etc.