Before we answer your question, you should keep in mind that many people use these terms interchangeably and will not make the distinction that we show below.
We will illustrate the terminology problem with the following example. Let's say your company is in the catering business and will cater its biggest event this evening. In the morning your company purchased about 125% of the paper goods that you believe will be used at the event. (You purchased the additional 25% for future events and also to ensure you don't run out of these items at this evening's event.) The paper goods that were purchased had a cost of $500, and only $400 of the paper items were used at today's event. The remaining $100 were put in your company's store room for use at the events to be catered in the next few weeks. In this example, the cost of $500 consisted of a $400 expense and a $100 asset.
Accountants use the term expense to mean a cost that has being used up while a company is doing its main revenue-generating activities. (That's why only $400 of the cost of supplies was expensed in our example.)
A cost may or may not be an expense. As we had seen above, $400 of the cost was an expense and $100 of the cost was an asset. Here is a more extreme example: If a company purchases land to be used in its business, the cost of the land will be reported an asset and will never become an expense. (The reason is that land will never be used up and therefore never depreciated.) The land's entire cost will continue to be reported on the balance sheet as the asset Land as long as the company owns the land.
If a company purchases a delivery truck to be used in its business, the truck's cost will initially be recorded as the asset Delivery Truck on the company's balance sheet. However, the truck's cost will become Depreciation Expense as the truck is "used up" in the company's main, revenue-generating activities.
Manufacturing companies offer additional insights. The cost of the direct labor used to manufacture a product is a cost of the product. If the product is not sold, the entire cost of the product (including the direct labor component) is reported as an asset. When the product is sold, the cost of the product that is sold will be reported as the expense known as the cost of goods sold (which is matched with sales) on the income statement. The wages paid to the driver of the manufacturer's delivery truck is reported immediately as an expense, since this cost is not part of a product's cost—it is part of selling the product. Electricity in the factory is part of the products' cost and will not become an expense until the products are sold. Electricity in the offices outside of the factory (sales and marketing, general administrative offices) are reported immediately as expenses in the accounting period that they are used. Cost outside of the factory do not become part of the products' cost.
Many authors of managerial and cost accounting textbooks add to the terminology problem when they say the break-even point occurs at the point where sales equal fixed costs plus variable costs. Perhaps there is some benefit in brevity, but I would prefer to say that the break-even point occurs at the point where revenues are equal to fixed costs and expenses and variable costs and expenses. There's a semantics problem with my preference as well.
I hope this sheds some light on the use of the words cost and expense. In 1985 the Financial Accounting Standards Board issued its Concepts Statement No. 6, Elements of Financial Statements, which defines terms such as expenses, losses. revenues, assets, etc. You can read that statement by going to www.fasb.org. In the panel on the left-side of the home page, select Pronouncements & EITF Abstracts.