The following images demonstrate the (minor?) difference that linear light processing makes when adjusting saturation. In both cases, the saturation has been increased 100% except in the flesh tone areas (where it has increased less: 30% only). *The saturation algorithm is Y'CbCr/YCbCr based.
In this case, linear light processing maintains constant luminance and the image does not appear to shift in brightness as much. Without linear light processing (i.e. staying in the image's original gamma-corrected color space), luma stays the same but luminance does not. The end result is that highly saturated colors become a little brighter (this effect is proportional to its original saturation).
Processed in gamma corrected color space (i.e. the original R'G'B'
Processed in linear light color space (gamma correction removed, calculations performed, then gamma correction added back in).
In my opinion, the image on the left looks a tad more pleasing since I prefer the brighter colors. However, one can blend between the two methods to generate other (and perhaps even better) results.
The image below is processed in gamma corrected color space. Roll-over to see the picture with linear light processing.
When an image is captured, it is typically converted into a gamma-corrected color space (such as Y'CbCr or R'G'B'). The input values have gamma correction applied to them. The Rec. 601 transfer function is:
f(x) = x0.45.
All the input values have a power function applied to them since human vision roughly follows an exponential progression.
To perform linear light processing:
 Other standards like Rec. 709 (used for most HD formats) and sRGB (for computer imagery) define slightly different and more complicated transfer functions than Rec. 601 (used for SD). Few conversions between Rec. 601 <--> Rec. 709 <--> sRGB take these minor differences into account; practically, the difference is too small to care about(?).
Rec. 601 and Rec. 709 also differ in their luma co-efficients as well as standard display chromaticities (the exact color of red, green, and blue; defined in chromaticity co-ordinates).
 Monitors will imperfectly reverse the transfer function that is applied during image capture. This is intentional. What goes in is not what comes out. Engineers found that making the output match the input does not result in the best-looking images. By playing around with the transfer functions, it is possible to somewhat compensate for glare, monitor brightness (most monitors are nowhere as bright as daylight scenes), and the viewing environment.