CI: Add public-api test

[joncinque]

Problem

#58 adds a step for cargo-semver-checks in CI, but that mainly checks for breaking changes. For more complete coverage, we can also check for any differences to the public API using a tool like cargo-public-api.

Proposed solution

Add a test as outlined at https://github.com/cargo-public-api/cargo-public-api?tab=readme-ov-file#-as-a-ci-check which checks for changes to the public api.

This might be quite noisy, so I'm not 100% sure about it.