If you went by sundial time, with the sun at its highest (noon) at 12:00, then the solstice, earliest sunrise and latest sunset would all happen within 24 hours of each other. But the sun appears to move east amongst the stars at varying rates throughout the year, meaning the time from one noon to the next is up to almost half a minute more or less than 24 hours. (See the other answers for why.)
Since we’ve had good enough clocks, we’ve preferred to pretend that noon to noon is always exactly 24 hours; that’s the average (“mean”) length of the day (hence Greenwich *Mean* Time). With this system, real noon can drift up to 16 minutes away from 12:00. It’s this drifting that throws off the dates of earliest sunrise and latest sunset.
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