How would a Bitcoin implementation of a TumbleBit payment hub compare with Monero?
What are the important differences between the two in the areas of privacy and security?
Update 2018.02.04 In light of recent of environment changes, this answer needs update, since blockchain scalability cannot be ignored anymore. TumbleBit's mixing technique became completely uneconomical, due to high fees. Although Monero has a similar fee problem, not as bad as TumbleBit has. That being said, TumbleBit has a Payment Hub mode, which is just as unusable, because it's unidirectional, however if someone figures out how to make it bi-directional, then things can change. As far as I know, nobody works on its payment hub mode.
You are specifically asking for privacy and security, so I will omit the practicality, where Monero would win by far if we wouldn't count that it needs to have its own token to work.
From a privacy point of view TumbleBit's payment hub might be slightly superior, since it's borrowing the anonymity set from Bitcoin users and if there are more Bitcoin users than Monero users that'd result in better privacy.
TumbeBit uses a central server, called the Tumbler.The only disadvantage having a central server is if the server goes offline no payment will happen, but this does not affect privacy.
From a security point of view there is not much to talk about. None of them has the risk of losing money. Both protocol is theoretically secure. How they are/ will be implemented is an other question.
I would also like to mention cryptographers might also make the argument that TumbleBit solely relies on text-book RSA cryptography.