Both specialists focus on oral health, but a periodontist is a more highly qualified specialist.
A hygienist is a mid-level medical professional (junior specialist, junior bachelor's degree). They are qualified to perform routine professional dental hygiene, educate patients about dental hygiene, and perform simple procedures (e.g., fluoridation). They can detect signs of inflammation but are not authorized to diagnose or treat.
A periodontist is a physician with a higher education (master's degree), completed an internship, and specialization in periodontology. A periodontist has a thorough understanding of anatomy, diseases, and treatment protocols. They are authorized to diagnose, prescribe and administer treatment, write prescriptions, and perform surgical procedures (gum plastic surgery, curettage, scaling, flap surgeries, etc.).