As with the other question,
A knows that no one 8 years younger can have the same birthday as his. On the face of it, all the century+41 dates mentioned in the answer to that question "work" because there was a king in the year in question.
That's not actually quite true, because
it wasn't until September 1752 that the UK switched to using the Gregorian calendar with its rule that most century-years aren't leap years. So 1441, 1541, 1641, 1741 are actually no good. But 1941 works fine and the UK had a king in that year, so perhaps that's our solution.
But, wait.
That switch to the Gregorian calendar meant that several days in September 1752 were skipped. So no one had 41st birthdays on any of those days in September 1793. George III was king during all this time. So the year could be 1793 instead of 1941. So it seems we have two solutions.
But (confession: I failed to spot this despite attempting to check; thanks to Jaap Scherphuis in comments for being less stupid than me)
the question asks us to spot the "differences", with the final "s" emphasized, for a reason: the only actual difference in the dialogue is the Queen->King change, but the preamble tells us (as it didn't last time around) that this is happening on a summer day and the birthday in question is the following month. That fits the September case (the gentlemen are meeting in August) and not the February case (it would be January).
So in fact
the year is 1793, the month is August, and our first gentleman was born in September 1744, somewhere between the 3rd to the 13th inclusive. The second gentleman cannot have had his birthday on the same day because it would have required him to be born in in 1752 between those dates, which were omitted from the year 1752 in the British Empire. (I think it is reasonable to suppose that a "stereotypical British gentleman" was in fact born in the British Empire.)
