A compound interest account pays interest based not just on the principal (your deposit), but any interest accrued and kept with the bank. Therefore, an account that pays compound interest will pay more over time than a simple interest account with the same interest rate. That difference becomes bigger the longer you keep your money in the account.
Even investments that don't pay compound interest can provide compound returns if managed strategically. Bonds, for example, pay simple interest. But if you use your interest income to purchase additional bonds, your returns have the opportunity to grow over the long haul. The same is true for noninterest-bearing investments such as stock mutual funds. If you use your fund distributions (if any) to buy more shares, you have the potential to achieve compound growth.