the level of integration with SSDT is way beyond that - the database structure and stored procs live side-by-side with your code, are source controlled together, built and deployed together. there's compile-time checks for stored procedures, code analysis, etc. DataGrip is just an alternative for SQL Developer/Management Studio, SSDT is much more.
It's also available standalone as "DataGrip", though I've never used it:
https://www.jetbrains.com/datagrip/