What's the difference between "academic staff" and "research staff"? For example in Cambridge website of math department, it lists University Academic Staff and Research Staff. What is the difference?
In the UK academic staff and research staff are typically used to circumvent funding regulations that limit many fellowships to individuals who do not have, nor have ever had, a permanent academic staff position. The fundamental distinction is that academic staff are on permanent contracts and research staff are on fixed-term contracts. Academic staff generally are on teaching and research contracts, but some can be on teaching only or research only contracts. I have never meet a member of research staff that wasn't on a research only contract.

Things get hazy because often research staff have two contracts. A fixed term research only contract and a permanent research and teaching contract dated the day the research only contract ends. If they then get a different fellowship, they tear up the research and teaching contract and get a new one with the new date. This allows them to continue to be research only and qualify for fellowships.

While there are obviously differences across institutions, looking at the provided pages for Cambridge, every member of academic staff I looked at had a academic rank (e.g., lecturer or professor) and non of the research staff did.

The tenure track is not really a thing in the UK anymore. Permanent contracts usually come with a 3 year probationary period. The requirements to pass the probationary period could be as little as do not screw up, but could also have publication and funding requirements. I am not particularly familiar with the publication requirements, but they are likely focused on the REF framework. Departments generally have a group For non-academic staff on fixed term contracts that have most of the same privileges as academic staff. Often this is the Research Staff group, but sometimes RAs and post docs get dumped into the research staff label. As I said above, these select non-academic staff people on fixed term contracts often have a permanent contract waiting for them if there is a lapse in their funding so for all intensive purposes they have tenure.