It's not so much that. Pikachu was never even when of my favorite Pokemon cuteness-wise (I am more of a Vaporeon and Dragonair man). It's the self-sacrifice Ash is putting himself under by giving up his friendship so that his friend may be happier, not knowing the tragic fact that it is not what Pikachu wants. And seeing Ash with the brim of his hat covering his tears and finally running away... :'(
That Pokemon battle each other is completely different. They do it for each other. If I cry around a battle sequence, it's because they lost and Ash is like "I'm sorry I've failed you Pikachu, and I'm sorry I've failed you mom and dad..." and I feel so bad for his self-deprecation. When he emerges victorious and Pikachu smiles knowing he is the most powerful Pikachu in all of Pokeland, it is a gloriously happy thing, even if he is covered in scrapes and worn down.