Legacy. Very long story short (it is still very long):
So if you want 1500 pixels by 1500 pixels, and have used illustrator pixel unit, you export at 72 dpi.
Now as to why the quality of illustrators export is not as good as in your monster image. Well, quite frankly Adobe isn't really very proactive when it comes to developing their tools. So they aren't using the best possible algorithm for anti aliasing their images. They are using a better one for the hardware accelerated screen, but you can not save using that algorithm to file.
Illustrator gives you 3 choices:
No Antialiasiation, which is obviously not good
Hinted AA (basically the way font work), which generally is too sharp
Art optimized, which is essentially quite good, it does a oversize render and box filtering. But obviously box filtering is not as good as you can get. But in addition adobe does not do color correct filtering.
We can do better naturally, personally i like using a lanczos filter. But this isnt really a done deal as people prefer different kinds of AA, and different kinds of AA on different devices. But here goes:
But your mileage may wary. Color correct aa tends to make lines thinner so its harder to design and no pixel preview makes it not so easy to use. That and having to do the filtering externally.
The reason you see the higer res image as looking better is that people making use of hardware acceleration usually get filtering right because hardware accelerator vendors have spent lot of money on this (read hundreds of millions). Since its unlikely that your image can fit your screen. Nothing to do with you DPI setting. Just doing different AA.