As far as I can tell, you gain nothing from doing these "quests" except 5 Gratitude Crystals and some changes to characters that doesn't seem to affect the story.
I tried both options and here's what happens for both:
(I forgot the exact details on some parts but I'll outline the decisions I made and what happened as best as I can)
My first run I gave it to the "voice coming from the bathroom."
The hand in the bathroom was grateful and claimed that it would not read the letter and would use it for "other purposes." Well, he/she/it lied and suddenly gets infatuated with Cawlin. I believe you get the 5 Gratitude Crystals here.
Telling Cawlin about it, he runs back to Groose's room hating Link for what happened. If you talk to him during the day, he will mention not being able to sleep. If you visit him at night, you will find the hand with him... doing stuff (nothing dirty ;). Karane just goes about her business. Nothing else happens throughout the game AFAIK.
Giving the letter to Karane, a number of things happened...
Before handing the letter to Karane, she mentions liking Pipit to herself. Proceeding to hand it to her, she asks if it came from Pipit. I chose to tell the truth and said it was from Cawlin. She reads it and is impressed but wants you to tell Pipit about this in the hopes of him having feelings for her.
Talking to Pipit, he contemplates whether he has feelings for her and I believe you have the choice to nudge him either way. I nudged him toward Karane and he decided that he loves her.
Returning back to Karane, Cawlin is there talking to her. He's there confessing his love for her. Pipit arrives and says that he's in love with her. (draaaamaaa) Karane thinks about what's going on and ultimately chooses Pipit. Cawlin runs back to Groose's room feeling sorry for himself. Karane and Pipit are together and you get 5 Gratitude Crystals from Karane. The hand disappears forever.
It looks like no path goes real well for Cawlin at all. ;)