夸张 is usually used as an adjective. Rose explained its meaning quite well. For example, 这个故事太夸张了。你的造型好夸张啊。这个效果很夸张。 夸大 is often used as a verb. If you 夸大 something, you describe or present it in a very dramatic way so as to impress or even cheat others. Your audience might feel your story is very 夸张, and you must be bragging about it.

They share a similar meaning. 夸张 means exaggeration, such as making something sound bigger or more dramatic that it really is. 夸大 also can mean exaggeration, but it has more of the connotation of someone is boasting what he is talking about.

夸张：通常就是用作形容词，或者是一种修辞手法。 夸大：一般情况下作为动词，夸大事实后就显得夸张了。

夸大是动词 夸张是形容词

夸大含有贬义的意思，例如请你不要夸大事实。夸张是一种说话或者写作的方法，例如中文古诗中有“白发三千丈”