The term 'apparition' is tied to early psychical research from the late 19th century in England. Originally a replacement term for 'phantasm', apparition was originally conceived to describe a visual event. Over time it became a general term for any apparent experience involving an unexpected encounter with a person animal or being of an apparent non-physical makeup. The term 'ghost' originally evolved in terms of 'haunting' apparitions. These are apparitions related to a place - many types of apparitions are related more to specific individuals. In modern terms, 'hauting' has (like ghost and apparition) become a more general term. But originally, the haunting apparition was solely an apparition of place and the things perceived in hauntings were called 'ghosts'.

In conversation with an English speaker, they will recognize both as synonyms. Alejandro's answer was spot on, but if you accidentally say, for instance, "the ghost of a cat" or "a human apparition" an English speaker will understand fine.

Ghosts are anthropomorphic. This means that they resemble humans. An apparition can be anything, and not necessarily humans. For example, cats, or demons.