навещать usually means visiting relatives (not necessarily old and sick, but it very often refers to elderly or sick) or visiting the sick in the hospital. Навещать usually implies that the person whom you visit is unable to come to your place (old, sick, has a baby, or similar circumstances). To visit smb's grave could also be translated as навестить могилу.

посещать is neutral. It just generally means 'to visit' (but more often places, like exhibitions, museums, gallery, might be a city or town, or classes and lectures; usually посещать means going to see something on purpose). Still, one can also use посещать referring to people. But with closer relatives навещать would be more common.

Навещать generally has a more personal connotation.