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A STARK and a SNARK are different proofs.
Ask what is hidden, what is posted, and whether someone had to generate a setup that must not leak. Do not ask which badge is more private.
The useful difference
A SNARK is often a short proof, and many constructions need a setup ceremony. If that setup is compromised, the proof can lie. A STARK is larger and is built from hashes, so there is no setup secret of that kind. Both can be used to compress execution. Neither checks the world outside the statement.