Right, quickly did an example. The original file is in Adobe RGB.
This version is saved with "Save_As" with the Adobe RGB profile embedded:

And this version of the same file is saved with "Save_For_Web" from the Adobe RGB file. Save for web strips the profile information. Then the browser assumes it is sRGB and the file looks washed out:

If you download the second one of these and open it up in Photoshop in an Adobe 98 RGB working colour space then it should look like the first file does in the browser.
Alex
p.s. I should add that in browsers that can't read colour profiles, both files will look washed out.