Simple interest and compound interest
Simple interest is the rate applied to the original amount, every period, forever. Ten thousand at five per cent earns five hundred in year one, five hundred in year two, and five hundred in year twenty. The interest never earns anything itself.
Compound interest adds each period's interest to the balance, so the next period earns on a larger amount. Year one is the same five hundred. Year twenty is not.
This page runs both on the same money so the gap is a figure rather than a slogan. Over a year the two are within a few cents of each other. Over five years the gap is noticeable. Over twenty it is larger than the original amount at many rates.
Where each one is actually used
Compound interest is the normal case for anything you hold: savings accounts, deposits, bonds where coupons are reinvested, and any investment measured by total return.
Simple interest is not a historical curiosity. It is how interest on many consumer instruments is calculated between payment dates, how statutory interest on late payments and on court judgments is usually defined, and how some fixed-term deposits that pay interest out rather than rolling it up behave. Anywhere the interest leaves the account rather than staying in it, simple interest is the right model.
That is the useful way to tell them apart: ask whether the interest stays where it was earned. If it does, compound. If it is paid away, simple.
The rate is what matters, not the compounding
It is tempting to treat compounding frequency as the lever. It is not. Going from yearly to monthly compounding is worth a small fraction of a percentage point of effective return; going from four per cent to five is worth a full point. Compare rates first and cycles second.
The other thing that outweighs the cycle is time. Doubling the term does far more than any change of compounding frequency at any rate this calculator accepts.