oh boy... here goes...
colour conversion depends on the conversion intent that you choose... the most commonly used in photoshop for photo colour conversion is "perceptual" and "relative colorimetric"...
taking the example to be aRGB to sRGB:
in perceptual, the entire colourspace of aRGB is squeezed into sRGB... as a result, the wider range of colour in aRGB is squeezed and this increase in colour contrast might result in puncher colours (contrast between colours) as represented in the image in question... the colours are not any more "real" or "brighter", they just seem that way because of this squeezing...
in relative colorimetric, the colourspace of aRGB is "overlayed" into sRGB, and those colours that are outside the possiblities of sRGB are squeezed into the sRGB gamut... only those beyond the sRGB gamut are squeezed, as opposed to perceptual, where the entire gamut is squeezed... there is thus the possibility that the colours beyond the edges of the sRGB gamut will be squeezed and be represented as similar to colours already in the image but which in aRGB might actually be different colours... thus, images could become less saturated in those areas after conversion to sRGB...
perceptual would be useful if the range of colour in an image is more important than absolute colour accuracy... relative colorimetric would be useful if the accuract of colour is more important, and we can live with, and possibly there is little, sqeezing of the colours at the edge of the gamut... you can choose the intent by going to "color setting", and at the "conversion options" area choose the one you want... try both out if an image is not satisfactory and see which one works better for an image...
so why should we clip the colours of aRGB before we post on the web? so we can make the decision of which type of conversion to use to maximise the potential of the image...