What is the difference between Partners, Principals, Directors at MBB consulting firms?

Principal - in practice the same as manager, but as this is the gateway position to the partner promotion, they sometimes do partner-level work. They have a stable salary with a variable bonus.
Partner - they sell the cases, typically have several assignments at any given time, and they oversee projects. Their involvement varies - sometimes you would see a partner daily on your project, sometimes you would see them once a month or even less frequently.

Director - a "super partner". They sometimes oversee junior partners, act as ultimate experts and sometimes they manage functional areas of expertise (e.g. Global head of Airlines Practice at Bain would typically be a Director).

Regarding salary, for principals it's a set salary with a bonus, for partners and directors the variable element is much higher. Although it depends on several factors, a partner can expect to make at least upwards of £150K plus bonus. There are rare cases when partners/directors earn about £1M and more. It is mainly driven by how much business they bring in.

Cheers!
Michal