These two words are too often used interchangeably, but they are completely different in meaning. "Imply" means to suggest or to indicate; "infer" involves a person actively applying deduction:
Water droplets accumulating on the outside of a cold glass of water can imply a hot humid day.
We can infer that Stonehenge was an early calendar.
Another way to look at it: We can substitute "suggest" for "imply" and "reason" for "infer," still retaining the correct meaning.