We all know from text books and practice that a knock out call is usually cheaper than a vanilla call option. Economically speaking, this comes from the fact that there is a probability bigger than zero of future spot price movements hitting the barrier. Consequently, the knock out option does not participate in every possible future spot price scenarios and that is why we do not have to pay the same premium as for a vanilla call.
If we now have a look at a Down-and-out Call with Barrier below strike, we see that the price matches a plain vanilla call option.
So, why should I buy a down and out call? Consider we hit the barrier below strike, then the option expires worthless. Consider the same case with a plain vanilla call. The option does not expire worthless and even has the possibility to participate on future spot upward movements. So why do we have to pay the same amount of premium for a down and out call as for a plain vanilla call?