From the link you provided, the main difference is (obviously) that DRI doesn't involve tone mapping. In a program like Photomatix Pro, HDR post-processing is divided into two main steps -- combining multiple exposures into a 32-bit file (no visual output), and then "Tone-mapping" that info into a smaller space. The painterly HDR "look" happens at this step, though there is a lot of flexibility with Photomatix Pro's tone mapper. Getting a realistic look through this technique is actually pretty tricky; maybe that's why they associate DRI with not using tone mapping.
DRI looks closer to "exposure blending"; the intended outcome seems a more natural rendering. At least the skies aren't overexposed.
As a new member, I can't post links, but a quick search for "HDR vs. DRI" will turn up a few articles, including a link to luminous-landscapes that describes a DRI process that sounds a lot like what people in other circles call exposure blending.