Aeroplane and airplane are different forms of the same word. Airplane is preferred in American and Canadian English, while aeroplane is traditionally preferred in non-North American varieties of English. But airplane has been steadily gaining ground in British publications, and it may someday become standard. Meanwhile, aeroplane is almost completely absent from American and Canadian publications, and to North Americans it may have an old-fashioned ring.
For example, these British and Australian news publications still use aeroplane at least some of the time:
Most other countries had opted to evacuate their nationals by aeroplane, many of which have arrived in Malta over the past two days. [Financial Times]
Yes, it is true that, like travelling on an aeroplane, the Oscars can be enhanced by upgrading to business class. [Sydney Morning Herald]
Another aeroplane woodcut print, it turns a newspaper shot of three military cargo planes on a runway into something awe-inspiring. [The Guardian]
And these Canadian and American publications use airplane:
The giant airplane maker has come up with a new interior for its 737 model, the workhorse of airlines around the world. [Wall Street Journal]
The money would likely be gathered through airlines and would show up on airplane tickets. [Globe and Mail]
Now, Alec Baldwin has been tossed off the air after his airplane incident apparently upset shoppers at a New York supermarket chain … [Los Angeles Times]
This ngram graphs the use or aeroplane and airplane in British books published from 1900 to 2008.