Ideally there should not be a difference. That is, different parts of model might be elaborated in more detail for test, others might be elaborated in more detail for development.
However, the scenario in which the test team goes off to make its own sequence diagram which is ignored by the development team is problematic to say the least. A recipe for frustration on both sides.
Of course, you may encounter a scenario in which the development team has not bought into the idea of model-driven development. In this case, making the sequence diagrams from the paper specification and asking the development team (who in this case has most likely developed little or no documentation of their own) to review them with you may be helpful.
However, the ultimate goal of model-driven development (or all sorts of systems - not just software) is to get all the stakeholders (test, electrical, mechanical, software, accounting, legal, etc...) to buy into the "model" as the "single version of the truth".