What's the difference between Staff and Senior engineer?

Senior engineer is usually focused on a single project, and knows it deeply. Usually directing and mentoring more junior engineers in the process.

Staff engineer typically is overseeing multiple projects, providing deep technical oversight and guidance on those projects, and mentoring Senior engineers. They start to influence technical culture. They are actively involved in ensuring business needs get met by the technical solutions their group is building.

Senior Staff Engineers will be overseeing a product function, and multiple Staff engineers. They build the correct technical culture. They ensure larger architectural issues get resolved. They are actively involved in ensuring the technical work being done is meeting business needs, and identifying business needs their technical org can be meeting - and working to make that happen.

Principal engineers are setting the tone for an entire large product (typically), and ensuring the Senior Staff engineers are doing the right things - and also often involved in driving strategic product direction.

Senior Staff and Principal tend to be increasingly political, but even Staff will get pulled into that type of thing somewhat regularly.