An expenditure is when you spend money, but receive nothing in return (credit card fee) while a cost is something you spend but you receive financial benefits in return.
Some good examples are:
Cost of goods sold = cost
Operating expense (i.e. SG&A) = Expense
Depreciation = Expense
This all being said, For case purposes I tried all of these in the "Costs" portion of my profit tree, and split more into FC and VC than Costs vs expenses. I'll lean on the other consultants here, but in my view you shouldn't really worry about this difference when solving a case (as it's an accounting concept, not a strategy concept).