whatever 是无论什么的意思, 是泛指所有东西的; 而whichever是无论哪一个的意思, 是有选择性的, 是指多个事物中无论哪一个,是在多个事物中选择。
例如:
He will give her whatever she wants. (她无论要什么他都会给她)
He has three houses, and will give her whichever she likes (他有3间房子, 她无论喜欢哪一间,他都会给她)