A down-and-in put becomes live only if the underlying trades below the barrier; an up-and-out call disappears if the underlying rises through it. Because the option may never activate, or may vanish, it costs less than its vanilla equivalent.
Barriers create discontinuous risk. Close to the barrier, the option's value and its delta can change violently on small moves in the underlying, which makes hedging difficult and can produce trading activity around the level as dealers manage exposure.
Most structured notes embed barriers, usually as a down-and-in put that hands the investor the downside once a level is breached. Whether the barrier is observed continuously or only at maturity materially changes the risk, and the distinction is buried in the term sheet.
Related: autocallable-note, reverse-convertible, structured-product, delta, put-option, implied-volatility