That story is about Fur Elise, not Moonlight Sonata. The name, of course, meaning "for Elise," the name of the woman in question.
The 3rd movement is not one of the hardest pieces to play. It’s like medium-hard. It’s not even the hardest sonata - that would be the Hammerklavier.
I'm not sure where I heard this, but Beethoven wrote this for one of his students that he was in love with. The first movement was purposefully written for a newcomer to be able to play--such as for whom he admired, but the third movement was a representation of his anger over being rejected, and is one of the hardest piano pieces to play.