The verdict
If anyone on your team writes code daily, I'd pick Cursor. It speeds up work in a real repository, and the team controls make it fit for a company. If you are a founder or operator who needs a prototype or an internal tool and has no developer, I'd pick Emergent, and I'd test the idea on the free plan first.
They can also work in sequence. Build the first version in Emergent, move the code to GitHub, and let a developer carry on in Cursor. Either way, a person reviews authentication, access rules and anything touching payments or personal data before real users see the result.