The way I describe the difference is that in 3/4 a single 3 bar measure is the complete rhythmic cycle for the piece. They are grouped together in the song of course, but each one is a complete unit.
In 6/8 you've got to hear all 6 beats to hear the complete rhythmic unit. Hearing only the first 3 beats doesn't give you all the information, because the second 3 beats are different somehow than the first three - which is why it's 6/8 not 3/4.
Let's say "ding" represents a cymbal hit or just a mental placeholder for a non emphasized beat; we're talking about the difference between "kick ding ding snare ding ding" (6/8) and "kick ding ding kick ding ding" (2 bars of 3/4). And every other part of the song is going to reflect the basic structural grouping in units of 6 rather than 3, although that is harder to explain or demonstrate concisely.
The difference between the two halves of the 6 beat cycle doesn't have to be that specific common one I just described, but there does have to be a difference .
If they were identical in every other way aside from the way they are counted/notated, there would be no need for 6/8 at all really. And that is the part of the explanation that is sometimes missing; it's not just a notation or counting difference.
It is sometimes described as being about "emphasis" or "accent" but I've always found those to be ambiguous. I can emphase rhythmically in all kinds of ways without breaking the basic rhythmic structure of a piece and so that's a difficult distinction to get across.
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