Why are there two different spellings depending on whether you are from the U.S.A. or Great Britain?

They're not merely different spellings, they're different words that mean the same thing.
As far as I know it's a historical difference between the English of North America and other English-speaking countries, like gasoline/petrol. I've never seen any record of either word being officially prescribed. Aerofoil/airfoil, aerodrome/airfield.

I think I'd call them different spellings, Can't be arsed to look it up but my guess is that when the word was coined (long before the Wright brothers) people used the normal Greco-Latin prefix aero- on both sides of the Atlantic, and at some point that was replaced by air- in the US in an effort at simplification. Although I guess you could say that that made the US word a compound noun rather than a prefix+root noun, arguably making them two words.
OP, if it surprises you that the Atlantic makes this much difference, how about the fact that on one side of the English channel it's aeroplane and on the other it's avion? And then cross the German border and it's Flugzeug.
Vinny since when can't you be arsed to look up a random fact. You sound a little up tight vinny. Perhaps you need a beer and a hot tub. a good sixth of the worlds problems could be solved with a beer and a hottub.

The difference in pronunciation between 'aeroplane' and 'airplane' is much less in US English than in (southern) British English, because the 'r' sound is retained in most varieties of US English. So perhaps the original coinage was 'aeroplane', and US English simplified it to 'airplane' without really noticing.
Aeroplane is the older word and actually predates fixed wing aircraft. It was coined in 1855 by a Frenchman
>In 1855, Joseph Pline proposed a dirigible glider, under, for the first time, the name "Aeroplane", in the shape of a bird, with a framework filled with gas, and fitted with propellers.
Because US citizens solve spelling difficulties in English by writing phonetically. (I bet they write fonetically, but maybe not.)
Which has nothing at all to with onomatopoeia used as an uncountable noun.

Then, there are aeroports, and airports, and planes, and jetliners, and jets, and airliners, and jet airliners, and jet aircraft, and aircraft,.. and at this stage, we are discussing language purity, and language prescription...
Here's a very American Aeroplane. Note the pronunciation.
Edited to add: such things really did exist
Edited by: Wilbur Wright