This workaround may help - Use asset export and output your files at 4x. Yes they are bigger, but they will look crisp. If you're outputting for the web and can use SVG, use that instead.

Two years later and i am still experiencing the same issues. (.png export for websites with 72ppi)
One tells me to export it as an .svg file, but that brings along alot of security issues.
The other says make it 300 or even 2000 ppi, but that makes the file format (00x00) and/or size (mb/kb) way to big. And i learned at school that you only use such high dpi should only be used for prints.
Another tells me its my retina screen or even just my eyes.
I have tried so many options yet they still all come out blurry...

****, fix this already.

In addition to my other comment, here's what a smaller export looks like. As you can see, the blur becomes even worse.

Update: after posting the image I notice this preview is displaying larger than it appears at 100% on my computer. Not sure if it's upscaling because the image was too small in dimension, but it seems this is not offering an accurate depiction of the level of blur I'm seeing on my end and what I'm trying to show.

Regardless, please know that the Adobe reps who have screen-shared and who I've provided screenshot vs export previews acknowledged the issue and agreed the level of blur is unacceptable. Just waiting for a fix and figured I'd validate what the poster of this bug fix request has mentioned.

I've been dealing with this problem for many months now as well. Adobe support likes to claim that we shouldn't expect raster graphics to appear as crisp as vector, though why am I able to take a screenshot of the vector file on my Mac and the resulting image looks nice and crisp?

I've had to resort to using these screenshots for my client's logo web files. Though since screenshots don't allow for background transparency, I'm unable to provide my client's with transparent website header png files (which of course is very unprofessional).

It's ridiculous that this issue continues to go ignored. I've reported it multiple times and have spoken with Adobe reps via screenshare and by phone, who each confirmed the level of blur to be unacceptable.

The level of blur is not "normal" and snapping artwork to a grid is not an effective or acceptable solution. Again, I'm able to achieve sharp png or jpg files via a screenshot, so why can't the world's top design software company achieve similar results?

As you can see by the attached images, the blur is very noticeable on a retina or high definition display.. The image quality of exports becomes even worse in smaller sizes, as the details become a complete blur. Yet no matter the size I can achieve a crisp raster image via a screenshot of the vector artwork. While screenshots have been my workaround, it is not a viable solution since I have no control over precise centering and can't achieve background transparency through screenshots.

I've tried every possible export setting and option from both Illustrator and Photoshop (as the 2 Adobe reps confirmed as they watched my workflow). So while we wait for a fix, what are we supposed to tell our paying clients who expect usable files for web?

p.s. After posting the images below I notice the screenshot looks a bit more blurry than on my screen, so I assume this forum downgrades quality some. Since I wasn't able to add a 3rd smaller image after posting, I added a smaller preview in a separate comment, showing how the level of blur becomes worse in small size.

It doesn't seem blurry to me, but I guess you have to snap you art to pixel grid.
Try 'Object > Make Pixel Perfect' command.