Outside of accounting terminology, bills and expenses are terms that are often used interchangeably. Technically, a bill is an expense. However, within accounting they do mean different things.
Bills refer to transactions that are planned and not yet paid for. These include services such as office rent, marketing services, etc.
Expenses refer to transactions that are created and paid for at the same time. Examples can include the cost of coffee at a business lunch or picking up a new cable for the office computer.