What benefits do sloths and manatees gain from algae growth on their fur/skin?
I don't know about manatees, but sloths have a symbiotic relationship with the moths living in their fur because the moths supplement the sloth's diet with additional nutrients via algae. Sloth moths grow and die in a sloth's fur, decomposing to provide nutrients for the algae. The sloth then eats the algae. What the moth gets out of this is that its larvae can feed on the sloth's poop, and when they grow up, they make a home in the sloth's fur again.

Here's an infograph summarizing the relationship: http://www.pbs.org/wnet/nature/sloth-named-velcro-sloth-moths-mutually-beneficial-relationship/11159/