Many tangs will eat it, but they can't live in the rather cold water off California. The Caulerpa, unlucky for us and every native critter, can tolerate it. The tarp-covering method used in CA to kill it kills everything else too because they actually have to inject chlorine under the tarps to do the job. The only other thing shown to be effective is hand removal with vacuum systems to suck up any broken pieces - successful in some Mediterranean areas but far too expensive for a cash-strapped state like ours to do.
Caulerpa also reproduce sexually, which can make them a big problem in aquariums. Yes, they grow fast absorbing lots of nutrients, but eventually they may decide to spawn - releasing tons of gametes which deplete oxygen at an incredible rate, and then the plants themselves die, further polluting the water. It can "nuke" a tank. Not all Caulerpa do this, though - it's suspected that CA and Mediterranean invasions of C. taxifola started as all-male plants, and only spread as "clones" via broken-off pieces. Meanwhile, C. racemosa has been observed spawning all over the Mediterranean.
Also, irresponsible aquarists are not the only source of this - the stuff can also spread via pieces attached to boat anchors and nets, and some think the gametes may survive in ships' ballast water.
What we all need to do is do our best to make sure the illegal species (those kinds that tend to be invasive and can survive in our local water temps.) are not in our tanks - and get rid of them very carefully if they are. (They can be brought in unknowingly, as hitchhikers on Caribbean or Gulf of Mexico live rock and corals. No one will ever intentionally ship you species C. taxifola in particular, as it is federally classified a "noxious weed" and all interstate transport is prohibited.)
Oh, and regarding roots - they don't have roots, but rather attachments called rhizoids. Unlike the true roots of things like seagrasses, these do not absorb mineral nutrients, hence are not roots - but they do look very much like actual roots, so it's easy to be confused.