In court, a party asserting that a patent is invalid must prove invalidity by "clear and convincing evidence," which is the highest standard of proof in U.S. civil law (that is, non-criminal law).
In contrast, suppose that the USPTO decides that a "substantial new question of patentability" exists and therefore initiates a reexamination proceeding. In that situation, the patent examiner need only show invalidity by "a preponderance of the evidence," which is colloquially summarized as 50.000001%, just enough to tip the balance of probability. It's also commonly phrased as "more likely than not."
(EDIT: Strictly speaking, the burden of proof applies only to facts, not to the ultimate conclusion. Here, the facts --- namely what the Hillis and Nomura prior-art references say --- are are essentially beyond dispute, although I suppose conceivably Apple might find some technical grounds to allege, as a matter of fact, that at the time in question, persons of ordinary skill supposedly would not have understood those references to mean A or B or C.)
Perhaps more importantly, in the court case, a non-expert judge and jury had to assess the prior art. In the USPTO reexamination proceeding, three experienced patent examiners did the assessment.