By you description, I'd guess that it's Gelidium, a red algae known commonly and red wire algae. Not a terrible problem, but it does creep across the rock like a creeping groundcover, and will gradually claim rock surface not occupied by coral or bushier macroalgae that can shadow it. IME, pacific Diadema sp. urchin eats it preferentially. Undfortunately, Diadema will also eat through the coraline algae that is under the Gelidium, scraping the rock to bare white. Oh well, it does really REMOVE the Gelidium, getting both the filaments and the holdfasts. Tuxedo urchins keep it at bay, but don't graze it once it develops into a healthy afro. I haven't encountered any other small consumers of Gelidium, but I don't know the grazing preferences of most surgeonfish, which can be grazers of tough alga. Hand harvest is of limited effectiveness, because you leave enough scrub for it to come back fairly quickly. However, in a small tank, hand harvest may be the best option. Or a small Diadema, which you pass on once it has beat the stuff back (and outgrown the tank).
I have a J. Sprung reference at home that talks about Gelidium. If it has better options than what I've given you, I'll post again.
