The short version: a monologue is any extended speech by one person, while a soliloquy is a specific kind of monologue, delivered when a character is alone on stage and speaks their thoughts aloud, usually so the audience hears what they really think.
Monologue (from Greek monologos, "speaking alone") is the broad term. A character may speak at length to other people on stage, as when Shakespeare's Antony addresses the Roman crowd in Julius Caesar, Act 3, Scene 2. A monologue can also be a stand-alone performance piece or a long speech in a novel or film.
Soliloquy (from Latin solus, "alone," and loqui, "to speak") is narrower. The speaker is alone on stage, or at least believes no one can hear, and the speech reveals inner conflict or reasoning. Hamlet's "To be, or not to be" (Hamlet, Act 3, Scene 1) is the classic case. Hamlet is alone, so it is a soliloquy, and it is also a monologue.
A handy test: if someone is being addressed or is listening, it is a monologue but not a soliloquy. If the character is alone and thinking aloud, it can be both.
Some scholars note that the terms were used loosely in early modern English, so Shakespeare's own texts label very little. The First Folio of 1623 does not mark most speeches with either word, so the labels are modern critical habits rather than his.
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