What is the difference between "激动" and "冲动"?
So for example, what is the difference between:
i would translate 激動 to "intensely emotional", where the emotion can be any sort of emotion, positive or negative (e.g. sorrow, anger, happiness). so it's very useful word because it can mean many things (despair, rage, ecstatic). and 衝動 means recklessness or impulsiveness driven by strong emotions, again any kind of emotion will do. typically the type of emotion is implied or from context.
when describing a person, 激動 is more of a mental state, where as 衝動 implies the person is physically getting ready for action (larger and more violent actions).
激動 examples： 他激動到哭出來。他激動到說不出話來。 (he is so emotional he starts to cry. he is so emotional he couldn't speak.) i used these examples intentionally to demonstrate that even though crying is an action, it's not big enough of an action so you can't use 衝動。
衝動 examples: 他一時衝動，把車窗打破。他衝動的跑去找上司理論。(he became reckless and broke the car window. he recklessly went to argue with the boss.) here the person is driven by the emotion (rage, displeasure or whatever) to do something reckless or impulsive.
衝動 can also be used to describe an action, where as 激動 only works on people. eg. 衝動的舉動，衝動的行為 (reckless actions).
so to answer your question, the first sentence implies the person should calm down in order to think clearly (or not have a mental break down), and the second sentence means the person should calm down and not do what he is about to do.
another usage you might find useful: 我這樣做會不會太衝動？ (would it be reckless if i did this?) for asking people's opinion when you feel like you're about to do something you might regret.
激动is more likely for psychological while
冲动is more likely for physical. For example:
得知自己的公司即将拿到新的投资了，我们几个创始人都很激动（Knowing that our company's gonna get a new round of investment, all the founders are very excited）
激动is more when something good happened while
冲动is more likely when something bad happened. For example:
他不是故意要害你的，千万别冲动啊（He's not offending you on purpose, just let it go）