You've heard someone say "aesthetic" as if the H were not there. They do this because they believe, with a certain amount of justice, that this is more "correct," or closer to the classical pronunciation. (They forget that one must disambiguate a theta from a tau in some way, though.) Fair enough. And the sequence "sth" is not the easiest to enunciate, so fair enough there, too.
But. I have never heard an English speaker do the same thing with either "anaesthetic" or "anaesthesia." Which I find interesting, because these words are from the same Greek root (αἴσθησις, if you're wondering).
Perhaps they feel that "anaesthesia" is a pedestrian word, connected only with such indelicate things as medicine, whereas "aesthetics" are things discussed over expensive wine in one's salon. Words that are insufficiently fancy do not deserve higher consideration, or a display of what they consider to be their elevated linguistic training.
But that's stupid, so I'm going to demand consistency.