An aeon is a stretch of time so long you cannot really feel it — far longer than a year, longer even than a lifetime stacked on a lifetime. Writers use it with a little drama: waiting felt like an aeon. Scientists use it more strictly. In the study of the Earth, an eon (the same word, usually spelled that way in American science writing) is the biggest slice of history; several eras fit inside one eon. The spelling aeon is the older, more British look; eon is the usual American one. Both feel a bit literary in everyday talk. People are more likely to say ages or forever unless they want the extra weight. It is not a unit you can count on a clock. If you are talking about a few weeks, aeon is a joke or a stretch, not a careful use.