What is the difference between soliloquy and a dramatic monologue?

The two are very similar, and I suppose they could essentially be considered to be the same thing. But as genres, there is a sort of difference. That is, a soliloquy can be thought of as a dramatic monologue - a monologue set within a drama. But a "dramatic monologue" (i.e. the genre) is a separate thing all of its own - a form of poem.

A soliloquy is something you'd find in a play, usually in a Renaissance play (and certainly one by Shakespeare!). It is a speech a character makes while alone on stage, usually to work out his thoughts. It's a bit like a sort of film-voiceover shot, where you hear the character speak but no-one within the world of the film can hear it.

So a soliloquy is a dramatic monologue: it's a self-addressed monologue within a drama.

But - a dramatic monologue is ALSO a specific Victorian style of poem. It's not within a drama but a poem all on its own, but written entirely in the voice of a single character (i.e. not a neutral speaker, nor the poet's own voice). Robert Browning is the poet most famous for using the form: the most famous examples being "My Last Duchess" and "Porphyria's Lover". Usually a character in a dramatic monologue will reveal more about themselves than they intend to - and a dramatic monologue usually contains rather a lot of irony! Unlike a soliloquy, there's also usually an "auditor" - someone listening to the character speaking the monologue.

Hope it helps!