Why do Wikipedia's and Guinness World Records's measurements of the Achaemenid Empire's percentage of world population differ?

Wikipedia is an encyclopedia with a firm no-first-hand-research policy; i.e., any fact contained in any article must be supported by a reliable source. Wikipedia itself (nor its contributors) has nothing to do with it.

So there can only be two reasons for those numbers in the Wikipedia being different from Guiness' numbers:

• Either the source (in this case, Scheidel, Walter (2020). "The Scale of Empire: Territory, Population, Distribution". as per the source listed on that Wikipedia page) has been quoted wrongly - i.e., a typo when transferring the data into the Wikipedia.
• Or Mr. Scheidel used other base data or heuristics than the folks at Guiness for whatever reasons.

Mr. Scheidel explains (a little bit) about his reasoning for his numbers; a footnote mentions that he uses "conservative corrections for the persistent underestimating of ancient population sizes" compared to his own, older source.

He goes on to state "it must be stressed that most of the numbers are merely rough approximations, best interpreted as the center values of wider ranges" , and "offer at least a sense of relative magnitude and broad patterns".

In that regard, I would suggest to view all this with an open mind - nobody has actual numbers for populations BCE, and it is clear that even where there are rare written traditions, numbers of general populations or armies, and so on, are often significantly inflated or deflated (i.e., the survivor of a conflict might inflate the enemy numbers, and deflate their own, to appear more heroic). Even assuming that the true numbers were actually known back then, with no general census or other reliable method to count them...

Guiness doesn't give a source, at least not in the article you linked, and I don't know their source policy.

If I were forced to decide between the one or other, I'd go with Scheidel's, purely because he seems to be relatively transparent about his source and procedure.

But the most important aspect is to be mindful that even the research paper is just that - scientific research. It is only a theory, and its truth value only consists in how long it takes to refute it. In this special case - ancient history - it is very close to being irrefutable (because we simply don't have any new methods to count the populations now, unless we invent a time machine), and thus is - same as 99% of other accounts of ancient history - just a tiny step above pseudo-science if taken as "absolute". (Don't get me wrong, articles like this are probably perfectly fine science if taken in the spirit of talking about the "history of history" - i.e. about the activity of doing historic research itself is the subject matter.)

Wikipedia gives population estimates for the Achaemenid Empire of between 17 and 35 million people (but without a year). Given that, at best, very few population records that may have existed at that time have survived, I am actually surprised the error margins are not even greater.

They give only one estimate for the world population in 500 B.C., which is 150 millions. But based on the different estimates given for 1000 B.C. and 200 B.C. one might conclude that 100 million would also be a reasonable estimate.

35 million divided by 100 million is 35%, and 17 million divided by 150 million is 11%. This would give a range which possibly might include the true value. The value from wikipedia is very close to the lower end of that range.

If we assume that estimates for world population and estimates for the population of the Persian empire are not independent from each other, it might be reasonable to divide the high estimate by the high estimate and the low estimate by the low estimate. This would give 35/150 = 23% and 17/100 = 17% as somewhat reasonable estimates.

Guiness uses a rather high estimate for the population of the Persian empire (49 million, possibly including some estimate for parts of Greece?) and a low-ish estimate for world population (112 million), which is how they arrive at their number.

Generally, it is a good idea to be sceptical about population numbers from more than, say, 200 years ago unless there is a very clear indication on what primary sources that number is based. Even today there are countries whose population numbers are not reliable at all (e.g. Afghanistan, which happens to be a former part of the Achaemenid empire).

The other thing you should be aware of is that relative error margins get larger when you multiply or divide. Above, the 34 million estimate is about twice as large as the 17 million estimate, and the 150 million estimate is 1.5 times as large as the 100 million estimate. But the 35% estimate is more than three times as large as the 11% estimate.