In Star Trek IV: The Voyage Home (1986), Spock becomes concerned by Kirk's sudden use of minor swear words but now, in the 25th century, even the "f" word appears among many others. This is merely due to the target audience and changes in accepted broadcast standards. Star Trek (1966): The Original Series was on TV, which rarely allowed swear words. The movies were aimed at family audience, so were restricted in what they could say.
If throwing an asteroid with a tractor beam was a weapon capable of damaging a starship with technology more advanced than possessed by Starfleet, it would long since have been adopted as a standard technique.
When speaking with Jack in the holodeck bar, Picard comments that he's about 23 or 24. It has already been established that Beverley dropped out of contact with Picard 20 years previously (roughly 2381). If Jack was indeed 23 or 24, this would mean he was born before the events of the film Star Trek: Nemesis (which took place in 2379) which is not possible.
Several Vulcans are shown hugging and caressing other crew members to comfort them, a very emotional act and very unlike a Vulcan.
When dictating his Admiral's log, Picard says the stardate is 78183. ten. Stardates are always stated as a numeral with one digit past the decimal, and since the number past the decimal is .10 there is no reason to state the second digit anyway.