The simplest way to differentiate the two is that antithesis is a contrast or opposition to something else. Usually, you have a thesis and the antithesis is the contrast or opposition to the thesis. A paradox is a self-contradiction, an oxymoron, or a word/phrase that signifies two contradictory meanings. Therefore, a paradox is like a thesis and an antithesis put together.
In philosophy or poetry, an antithesis can be used to oppose a first (thesis) proposition. In William Blake's "The Tyger" (Tiger), the speaker asks who could make such a ferocious animal. The speaker is perplexed as to who (presumably asking God) could make a ferocious thing such as a tiger but also make something as docile as a lamb. Lamb is the antithesis of the tiger. Analogies of this thesis/antithesis could be: tiger/lamb, violence/peace, predator/prey, etc.
Did he who made the Lamb make thee?
Tyger! Tyger burning bright
In the forests of the night,
What immortal hand or eye
Dare frame thy fearful symmetry?
An example of a paradox from Hamlet is when Polonius says that Hamlet is acting both mad (crazy) and methodical (logical).
Though this be madness, yet there is method
Polonius recognizes that Hamlet is behaving like a madman but seems to have some reason or rationale for behaving so. To behave logically and like a crazy person is a paradox. This paradox is resolved when the reader figures out that Hamlet is mostly acting mad to disguise his strategy for revenge. But there will be times in poetry when a paradox is never resolved; this might be to state a duplicity about a character or a paradox could stay unresolved to retain a double-meaning.