On Windows it isn't possible to remove or rename files and directories that are being used. In order to handle this, you typically have to run some sort of helper application outside of the installation directory. This renames the existing installation directory to a new directory with ".tmp" (often, not always, but this is my guess). If that rename operation fails in the middle of the operation because a file being opened, either by another process or sometimes by an add-on, then the background update fails.
On Mac and Linux it's less complicated which is why they never get the "download the new version" prompt, but Chrome and Firefox have evolved systems for managing how these silent updates work. On Chrome the update button you talk talking about means you haven't closed the browser in awhile. If you see green that means an update has been available for 2 days, orange – 4 days and red – 7 or more days.
My guess is that you are encountering a situation in which the Chrome background helper can't successfully run the update process and then asks you to download the whole thing again so they don't need to rely on the updater and can just move your user files over.