Cost is used on something that has returns, while expenses are expenditures used on things that depreciate.
Cost is reported in the balance sheet because it means more funds will come to the balance sheet after the expenditure. Expense on the other hand, is reported on the income sheet because it will be taking away funds from the income sheet in making the expenditures.
Cost is used when one is buying assets while expense is used on buying liabilities or things that eventually expire.
Unfortunately, cost and expense tend to be used interchangeably even within the accounting terminology. Even the "cost of goods sold" term just used should really be called "expense of goods sold," since it refers to costs that were charged to expense when a sale of goods was completed.