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.
冲动 means to be inclined to do something suddenly , regardless of its outcome which usually is dangerous and harmful.
The meaning of 激动 is relatively neutral.
The best way to translate these two is 激動 as excited, and 衝動 as impulse or drive.