Translation guide
The medical specialty focused on health care of elderly people.
The field of medicine dealing with the health and care of older adults.
The hospital department or division that provides geriatric care.
Medical care for the elderly, not necessarily as a formal specialty.
老年医学 is the more formal, academic term. 老人医学 is also used but can feel slightly more colloquial or patient-focused. In most contexts, 老年医学 is preferred for the specialty.
When talking about geriatric medicine in everyday conversation, Japanese speakers might simply say お年寄りの医療 (medical care for the elderly) rather than the formal term.