Saathane Square is the closest Samsun comes to an old civic heart. The clock tower, mosque, small shops, and market streets give you a rougher, more local counterpoint to the seafront and the official Ataturk sites.
Come during the day, when the surrounding streets are busy with errands and food shopping. It is not polished, but it is useful if you want to see Samsun as a lived-in Black Sea city rather than a line of museums.