Deriving one from the other
A half-life of 10 implies a per-unit-time decay rate of about 6.6967% — found by solving (1 − rate) = (1/2)^(1/10). Feeding that real decay rate, an initial value of 100, and 30 periods into the general exponential decay calculator gives a final value of exactly 12.5. The half-life calculator, given the same 100 units, a half-life of 10, and 30 units of elapsed time, independently reports the identical 12.5 remaining.
Why the two calculations necessarily agree
A half-life of 10 and a per-period decay rate of 6.6967% aren't two different decay processes — they're two different descriptions of the exact same one. Converting between them just re-parameterizes the identical exponential curve, so any point computed on that curve has to come out the same regardless of which description was used to get there.
Why half-life is the more natural description for many real quantities
Radioactive isotopes, drug elimination, and similar processes are conventionally described by their half-life — a single memorable number — rather than an awkward per-unit-time percentage like 6.6967%. The decay-rate formulation is more natural when a process is instead described directly as "shrinks by X% each period," like a declining-balance depreciation schedule.
The practical value of converting between them
Being able to move between a half-life and its equivalent decay rate means either description can be plugged into whichever calculator is more convenient for the rest of a problem — useful when combining a half-life-described process with other quantities already expressed as period-by-period percentage rates.