Fee v. Cost

Fee - an amount of money paid for a particular piece of work or for a particular right or service.

Examples:
How much is the registration fee?
We have added the doctor's fee to the bill.

Cost - the amount of money needed to buy, do or make something; the amount of money needed for a business or to do a particular job

Examples:
I wish I could install this software to my new laptop at no extra cost (that is, for no additional money).
Management told the marketing people to cut advertising costs.

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