Software engineers work with direction - given a project plan and architecture, they can write code that works and get it into production.

Senior software engineers work without direction - given a business requirement, they can scope work and develop a plan that will get code into production.

Staff software engineers work across team by providing high level requirements and direction, complete enough for senior engineers to scope it and turn it into a project that eventually gets code to production. Instead of taking a simple project and executing, they're probably / typically helping expose tradeoffs in the projects within a line of business / department, and helping prioritize features that make future work easier.

Principals are somewhat like Staff, but instead of working for one line of business, they cross LOB and work to ensure the company is both building the right thing and building it in the right way.