The difference between a waxing crescent and a waning gibbous is primarily their phase in the lunar cycle:
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A waxing crescent is the phase of the moon when less than half of the moon's surface is illuminated, and the illuminated part is increasing. This occurs after the new moon and before the first quarter.
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A waning gibbous is the phase when more than half of the moon's surface is illuminated, and the illuminated part is decreasing. This occurs after the full moon and before the last quarter.
In summary, the waxing crescent is increasing in illumination, while the waning gibbous is decreasing.