Deliberate mistake: When Apollo lands the Cylon fighter aboard Galactica, the view forward from behind he and Starbuck is a reuse of the same shot from the prior episode when they landed a shuttle aboard Celestra. Note the headphones they wear, then don't wear in the very next shot.
Factual error: Apollo's monitor picks up the "Tranquility Base here: the Eagle has landed" message of the first moon landing, complete with a full shot of the LEM sitting on the moon's surface - not from the onboard or outboard cameras, but from several yards away. There's absolutely nothing to indicate that this is some edited rebroadcast (that's one stretch of a rationalization). The implication is clearly that it's live. And when the Eagle first landed there were, of course, no cameras on the moon. (Nor is it necessary to insist that the series would have to be set in the Earth-time 1960s for this to be so. A live signal could take many, many years to reach the Galactica.)
Factual error: The Cylon commander orders his centurions to "continue into the galaxy." This is roughly tantamount to the GPS in your car instructing you to "continue to the nearest planet." It makes no sense, as they're not entering any other galaxy (nor could they). Yet again, the writer obviously confused his terminologies, and should have written "system," not "galaxy." Speculating that he meant this galaxy is just silly; it's clearly not what was intended.