Comment by Spivak

You have it right. The problem Anubis is intended to solve isn't bots per se, the problem is that bot networks have figured out how to bypass rate limits by sending requests from newly minted, sometimes residential, ip addresses/ranges for each request. Anubis tries to help somewhat by making each (client, address) perform a proof-of-work. For normal users this should be an infrequent inconvenience but for those bot networks they have to do it every time. And if they solve the challenge and keep the cookie then the server "has them" so to speak and can apply ip rate limits normally.