Consider the case where a user is submitting an e-commerce order. You want to mark their order as processed and submit it for fulfillment. If you read the order to check if it's already submitted, two requests to submit it made at almost the same time (e.g., hitting the button twice) will both read that it's unfulfilled and try to each submit it.

By doing an atomic write you can be sure that at most one request submits the order.