Opponent process is a color theory that states that the human visual system interprets information about color by processing signals from cones and rods in an antagonistic manner (source).
What is the advantage of opponent color against RGB color? For example, in object recognition or edge detection?
Can we say that it provides some level of invariance to changes in brightness?
RGB color is a more direct representation of the "raw input" received by the (human) eye, since the three types of cone cell have responsivity spectra that roughly correspond to red, green, and blue light:
Opponent color decomposes color into three "dimensions" of opposing properties: dark or light, yellow or blue, and red or green. This provides some brightness invariance, since changing the intensity of light will change the brightness but not the yellow/blue or red/green balance of the color, even though the stimulation level of all three types of cone cell will change.
One could speculate that opponent color perception is more "meaningful" in real-world environments. The dark/light axis represents the overall amount of light, the yellow/blue axis indicates which end of the visible spectrum is predominant, and the red/green axis provides a finer distinction for low-frequency light.
The first question, namely what is the advantage of color opponency 'against' RGB, is technically incorrect. The opponent system (Red/Green; Blue/Yellow and brightness channels) is physiologically situated in the neuronal retinal layer and higher visual structures such as the lateral geniculate nucleus and receives input from the RGB system (photoreceptors in the retina: Red, Green, Blue cones).
Then the question becomes "what is the advantage of adding the opponent channels on top of the RGB system? The advantage of opponency may in fact not be the negative opponent effect so often stressed in the Hering model (and shown in the figure: e.g., blue suppressing yellow etc). The advantage lies in the finding that activity of one opponent pair increases activity of the opponent color in adjacent areas in the retina. This effect results in a sharpening of color contrasts between the opponent colors (Hurvich and Jameson, 1957). Indeed, the poor spatial color-contrast, as provided by the sparsely placed cones in the retina, is enhanced by this mechanism. Especially the blue cones are present in isolated patches in the retina. Because one color in an opponent pair increases the other's activity directly adjacent to it in the retina, the resolution of the blurry retinal image is greatly increased. Hence, as you rightfully say, edge detection and in particular color contrast perception is enhanced.
Object recognition, as you mention in your question, is a process that is facilitated by the visual system through many stages including the higher visual cortices. Enhanced color contrast will likely facilitate object perception, but indirectly since color contrasting makes up just a small portion of a much bigger and complicated neural system devoted to object recognition, namely the visual 'ventral' stream (Ishai et al., 1999).
Lastly you ask whether opponent colors may facilitate invariance to brightness. Assuming you mean color invariance under varying luminance (referred to as color constancy) the answer is no. The RGB cone system can deal with this easily, as the ratio of cone activation yields enough information to code the hue under various luminances. If a certain object reflection activates twice as many red cones than green cones, it will also activate the same ratio when the luminance is increased (Mather, 2006 - Chapter 12).
