Robin's final speech at the end of the play is both a monologue and perhaps, a solo address.

A monologue is a single person speaking alone, an audience is optional. Puck is speaking alone; there are not other characters that join him. He also is speaking to an audience. 

Puck's last lines may also be a solo address. A solo address is one of the four types of monologue. 

A solo address is an explanation to the audience. At the end, Puck addresses the audience. He tells the audience what he is doing and asks for the audience's forgiveness.

A monologue is a speech given by a single actor, and a soliloquy is the same thing, except the character is alone on stage, speaking to him/herself. An aside is a line spoken directly to the audience that other actors/characters cannot or do not hear.

I would consider Puck's speech a monologue. It's not a soliloquy because he's not speaking to himself or speaking his thoughts out loud. Though he is speaking to the audience, it's not an aside, as it is a direct address to the audience. Technically, it's an epilogue or a curtain speech, as Puck is asking the audience to like the play and if not, pretend it was all just a dream. It's a common convention in Shakespeare plays...look at "As You Like It".