I would really appreciate it if you could explain the difference between the two nouns. Which one has to do with physical sensations and which with mental abilities? It would also really helpful if you could provide an example.
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I would say that in the most general terms, all emotions are feelings, but not all feelings are emotions.
I'm feeling tired (lucky, lazy, sick, dizzy) tonight. I've got a suspicious feeling about him. I've got a strange feeling about this presidential candidate. These are not emotions.
On the other hand, since emotions (e.g. love, anger, hatred, worry) are things you feel, emotions are by nature a type of feeling.