Difference between $1-P(A)$ and $P(A')$ in probability

The difference is that you're failing to account for the possibility that a disk fails exactly two or three tests, so you're (slightly) undercounting the failure rate.

The correct probability for at least one to fail is, by inclusion-exclusion, $$P(A)+P(B)+P(C)+P(D)\\-P(A\cap B)-P(A\cap C)-P(A\cap D)-P(B\cap C)-P(B\cap D)-P(C\cap D)\\+P(A\cap B\cap C)+P(A\cap B\cap D)+P(A\cap C\cap D)+P(B\cap C\cap D)\\ -P(A\cap B\cap C\cap D).$$ Assuming the events are independent, this is exactly what you would get if you multiplied out $(1-P(A))(1-P(B))(1-P(C))(1-P(D))$ and subtracted it from $1$.