1. 表示“担心的”、“焦虑的”这一意义时，只能用 anxious，不用 eager；表示“为……担心”，一般后接介词 about, for。如：

We’re anxious about [for] her health. 我们担心她的健康。

He was anxious for his family, who were travelling abroad. 他担心在国外旅行的家人。

2. 表示“热切的”、“渴望的”这一意义时，两者都可用，但有以下几点需注意：

(1) anxious 之后一般用介词 for，而 eager 之后可接介词 for, after, about 等。如：

I’m anxious for a computer. 我渴望有台电脑。

We are all eager for [after] knowledge. 我们都求知心切。

(2) 两者之后均可接“(for sb)＋不定式”，表示“渴望或想要(某人)做某事”。如：

We are anxious [eager] to know the result. 我们渴望知道结果。

He was anxious [eager] for her to go with him. 他渴望她 同他一起去。

(3) 两者之后均可接 that 从句，且从句谓语一般都用“should＋动词原形”这样的虚拟语气。如：

He’s anxious [eager] that she should come to see him. 他希望她来看他。

注：上面也可改成：

He is anxious [eager] for her to come to see him.