In his monologue, Claudius summarizes the latest happenings in court and tries to cause people to believe that his brother's death, his following coronation and his marriage to his brother's wife in quite a short time are natural consequences that should be ignored quickly.
A soliloquy is a device for characters to actively exhibit their emotions and ideas in a loud voice, so that only the spectators can hear - not the rest of the characters. Hamlet's soliloquy shows his feelings, thoughts and distrust towards Claudius.
