How would you describe the subtle distinction between sensation (feeling) and emotion? Why did the Buddha teach that it is so important to learn to observe sensation?

I would have to say that to me, a sensation precedes emotion. Say I feel something that causes me discomfort - whether it's a headache, a sore muscle, or even a heartache. That's a physiological sensation. Once my brain notices these physiological sensations, it interprets this sensation based on what is going on in my environment. So if it's a headache, I may interpret the throbbing pain in my head is related to stress, anger, sickness, grief, etc. If it's a sore muscle, I may conclude that I am feeling discomfort in my muscles because I am being held too long in a yoga posture. If I feel a sort of pain or hallowness in my heart I might attribute this to sadness, emptiness, anger, etc. To me I think emotions are responses to physiological stimuli. I could be way wrong though. But this is how it all makes sense to me.

The Buddha says we need to learn to observe sensation because if we don't observe the sensation for what it is (a physiological response) our minds can interpret this response and let it turn into something bigger than what it really is. Considering the muscle soreness from holding a posture longer than I normally do. If I don't know that the discomfort is just discomfort, my mind will tell me it's the instructor's fault for keeping me here this long. My mind will then tell me that they're an awful teacher who likes to do nothing but torture their students, and go on and on, possibly psychoanalyzing said teacher as a control freak, linking them to all sorts of behavioural and personality disorders, and so on, and so forth. That is the reason we need to keenly observe sensation - if we don't we end up allowing our emotions to control us all of the time.