They have different hierarchies. In typical 3/4, the first beat of each measure is given equal weight. In typical 6/8, the first beat is given a bit more weight than the fourth beat.

Same reason why we treat 2/4 and 4/4 differently.

Who's we?

Your decision to give certain beats more or less emphasis is a subjective one. You can glean absolutely no information about a piece from the way the rhythm is subdivided, except maybe the intended tempo.

Perhaps you work in an idiom that has no connection to the conventions of the notation system…?

So, for example, lots of pop/rock music gets notated rather arbitrarily in practice because it's mostly about the feel from recordings anyway. It's common in that world to see what classical convention would call incorrect notation. And since the core notes all work still and you can go by the feel from the sound you know, it doesn't really matter.

But the classical conventions include ideas that the subjective accents you describe are in fact implied by certain time-signatures.

2/4 v. 4/4 is more about the number of quarter notes (beats) per measure (2 v. 4, respectively).

Similar deal with 3/4 v. 6/8. In 3/4, you have 3 beats per measure, subdivided into halves (duplets). In 6/8, you have 2 beats per measure, subdivided into thirds (triplets).

The whole point of the original post and of the lesson describing 3/4 vs 6/8 was not to compare those two notations that indeed are effectively interchangeable.

The point was to compare (1) & (2) & (3) & to the different accent pattern of (1) 2 3 (4) 5 6.

Anyway, even in the case where you are instead counting the 8ths of 6/8 with the same timing as the quarters of 3/4, the 3/4 more readily allows the possibility of odd-numbers of measures in phrases.

That's like how 2/4 and 4/4 are truly identical if the notes are the same (the same son in either time signature is unchanged), but only in 2/4 can you readily have 5-measure sections (i.e. 10 quarter notes) as phrases.

Anyway, there's obviously some miscommunication between us. I assume you actually would agree with everything I'm saying if we were discussing in person. But your original post here will read to many people here as making the claim that there's no distinction between:

3/4 A E C E C E (all 8ths)

and

6/8 A E C E C E (all 8ths)

whereas the whole point of the original article was to describe that these identical timings have different feel because of the different accent structure.