These are mostly non-specific terms.

"Ancient" is used mainly to refer to larger areas that are old, especially when the civilization in question was organized differently in the past from today, as in Ancient Greece, Ancient Egypt, Ancient Israel. It usually is used when the ancient name for the area is the same as the modern English name. There really is no specific time period that "ancient" refers to, just that it's pretty old and was in a different system back then, in order to separate it from the modern entity of the same name.

"Classical" refers to the Greek and Roman sphere. Anything to do with ancient Greece and Rome, basically between maybe 500 BC and 300 AD.

"Antiquity" can have a couple meanings. When used referring to a time (as in "in antiquity") it really just means "in the past". Usually referring to the same time periods as "ancient".

An antiquity is an artifact of an ancient period.

In academia, time periods are referred to by their names, for the sake of accuracy and precision. Rather than talking about Ancient Israel, we might talk about the Iron Age IIa (roughly 1000-925 BC), which is a different time period than the Iron Age IIb-c (925-586 bc), because the culture and organization was very different in those times. Similarly, when dealing with the Classical periods, a scholar might refer to the Severan period of Roman emperors, the Late Roman, etc. The three terms you are describing are very general, and are used in a wide variety of situations by scholars.