Monologue or soliloquy?
Monologue: noun - long speech made by one person, either in company or conversation; discourse in the nature of a soliloquy; scene in a drama in which a person talks alone, a dramatic composition for one voice; a dramatic entertainment performed by one person; form of dramatic scenes or compositions for one person; literary composition in the form of a soliloquy.
Soliloquy: noun - literary representation of an instance of talking to yourself regardless of an audience, part of a play involving this; action of making a soliloquy or monologue.
As is so often the case in English, there is little actual difference between these terms, and any variations are subtle. It comes down to a matter of taste and, as always, consistency in choice is perhaps more important than any other consideration. However, I’d tend to use ‘monologue’ for something comic or plain and ‘soliloquy’ for something literary or flowery.
So, I’d label the famous deliveries given by Stanley Holloway (The Lion & Albert), and I’d call the famous speech by Hamlet a ‘soliloquy’ (To be or not to be).
Pic: Looking south east along Deep Dale