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Postgres’ long-term native support for ACID compliance, Multiversion Concurrency Control (MVCC), data and referential integrity, server side programs (triggers and stored procedures), and fine-grained access controls are much more mature and stable than MySQL, which only has started implementing some of these features in recent years.
While not an exhaustive side-by-side comparison, this whitepaper covers a few key differences between Postgres and MySQL and some of the benefits data professionals will find in working with Postgres.