"be annoyed with"表示对某人感到烦恼或恼怒。而"be annoyed at"表示对某件事情感到烦恼或恼怒。它们在释义、用法、使用环境、影响范围和形象上存在一些区别，具体内容如下所示。
概述：
1. 释义区别：
"Be annoyed with"意味着对某人感到烦恼、不满或不快，并且通常与某人相关。
"Be annoyed at"意味着对某事感到烦恼、不满或不快，并且通常与某事情相关，而不仅仅是与某人有关。
例句：
I am annoyed with my brother for always borrowing my things without asking.
（我对我弟弟总是未经允许就借我的东西感到烦恼。）
She was annoyed at the late arrival of the train.
（她对火车的迟到感到恼火。）
2. 用法区别：
"Be annoyed with"通常后接名词或代词作宾语，表示针对某人或某人的行为感到烦恼。
"Be annoyed at"后面通常接名词、代词或动名词，表示对某个具体事物或情况感到烦恼。
例句：
He is annoyed with his friend's constant complaining.
（他对他朋友不停抱怨感到烦恼。）
She was annoyed at the noise coming from the construction site.
（她对建筑工地传来的噪音感到恼火。）
3. 使用环境区别：
"Be annoyed with"常用于个人关系或亲密关系中，表示对某人的行为或态度不满。
"Be annoyed at"更常见于客观情况、具体事件或公共场合中的怨念和烦恼。
例句：
The teacher is annoyed with his students' lack of attention in class.
（老师对学生上课时缺乏注意力感到烦恼。）
I am annoyed at the constant traffic jams during rush hour.
（我对高峰期不断发生的交通堵塞感到恼火。）
4. 影响范围区别：
"Be annoyed with"强调情绪主导，更多地反映个体内心中的烦恼。
"Be annoyed at"则指向外界因素引起的烦恼，更多侧重于特定事件或情况。
例句：
He is annoyed with the constant interruptions while trying to work.
（他试图工作时不断被打断让他很烦恼。）
She was annoyed at the poor customer service she received at the restaurant.
（她对餐厅提供的差劲客户服务感到恼火。）
5. 形象区别：
"Be annoyed with"常被解释为一种较为私人的情感，带有更多的抱怨或反感的情绪。
"Be annoyed at"则更倾向于描述因某个特定触发事件而引起的愤怒、恼怒或不满的情绪。
例句：
He is annoyed with his sister's constant nagging.
（他对姐姐无休止的唠叨感到烦恼。）
She was annoyed at the unfair treatment she received from her boss.
（她对老板给予她的不公平待遇感到恼火。）