What is the difference
With simple interest, the rate always applies to the initial capital, so the earnings are the same every month. With compound interest, the rate applies to the accumulated balance, including the interest of previous months ("interest on interest"), and the earnings grow over time.
Simple interest: M = C × (1 + i × n)
Compound interest: M = C × (1 + i)^n
Example with $10,000 at 1% per month for 5 years
With simple interest, 60 months at 1% earn 60% on the capital and the total is $16,000.00. With compound interest, the total is $18,166.97, $2,166.97 more. The difference is small at first and grows every month.
Which regime applies in practice
Fixed-income investments and most credit card debts and loans work with compound interest. Simple interest shows up in some contracts and specific calculations. Always check the terms of your contract or product.