As with so much of this, I think the best answer is "it depends."
Depends on the coffee, the filter, the water, the preparation method, etc. But, at the risk of over-generalizing (and comparing the Chemex only to my Hario):
* the Chemex makes a larger pot (obviously), depending on the size of your Chemex.
* the Chemex tends to make a "smoother" brew than the Hario, I think because the Chemex uses thicker filters. You can, of course, find alternate filters for sale. The Hario makes coffee that is, well, whatever the opposite of "smooth" is. I would put it this way: whatever the coffee "is," the Hario will make more of "that" than the Chemex. So, if the coffee tends to the bitter, the Hario brew will be more bitter than the Chemex. If the coffee tends to be "robust," the Hario will brew a more "robust" coffee than the Chemex.
* if you subscribe to the "religion of the importance of the pour," it's easier to impact the taste on the Hario with the pour vs. the Chemex.
I like the Chemex went I want a "smooth" cup to savor over time, over a long evening.
I don't know. Others with more experience can probably explain this better than I can.
Each person's coffee experience is personal to them. I obsess over the size of my post-grind grains because doing that enhances the coffee experience for me, both in terms of taste and other parts of the experience. Let's say there is a scale out there called "how much person X just enjoyed that cup & experience." I wonder. Maybe on that scale I would be an 8, this morning. But, who is to say that someone who this morning drank that Folgers I linked above did not hit a 9 on that same scale?
In other words, sometimes I think I do all of this prancing around and what-some-think-is-snobbery-type stuff (but I hope I never come off that way, it really is not) just because I enjoy doing it. (Speaking only for myself, of course.) I would rather drink an over-roasted, burned Starbucks cup-of-crap with a good friend than drink the perfect espresso alone.
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ETA: I use the Aeropress 10x more than the Chemex or the Hario. And I use the various espresso methods 50x more than I use the Aeropress.