As far as I can tell, CC adds a few extra X-embeds inside a .toc file, which, while not causing any problems ingame, can be considered by another updater as an "alteration". Your best bet imo, is to do what Phanx suggested :
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...go through each addon that appears on both sites, figure out which on the author updates most frequently (or provides more prompt support, or whatever) and tell the other site's client to ignore the addon.
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Simply put, you can run both updaters just fine, as long as you configure which one will be handling updates for duplicate addons (just to be on the safe side).