There are so many, many advantages.
Dolphin is about as fast as Mini and faster than stock, but you can load the full featured pages for sites that redirect you to mobile pages (that totally suck and lack content). Even the stock browser set up as Desktop still doesn't load, say, cracked.com, only m.cracked.com.
Unlike Mini, Dolphin supports multi-touch. Like Mini, it reformats text into columns for easier reading. Unlike Mini, it remaps the volume rockers into pg up pg dn keys so you can scroll through the pages.
Dolphin has gesture controls that are shortcuts for commands or webpages. This feature is functional and clever by a half.
Mini has this bad habit of reloading the page instead of loading the linked page.
Dolphin will display placeholders for flash content and depending on the site,will send the URL for the video to either HTC flash player or the YouTube app if you click on it (your choice). So it doesn't support flash natively, but then again, you don't take the hit when you go to a site with lots of flash (the stock browser with flash enabled slows to a crawl when that happens). In Mini, there's just blank space.
Basically, it's the best mobile browser on the planet. Nothing even comes close. The only complaint I have is the chunky and barebones bookmarking system. But I can map a lot of my frequently visited pages to gestures.
Oh, and it can take over as the default browser for the system. I'm not sure if Mini can do that in Android -- it could not in WinMo without hacking the registry (and even that didn't really work).
Like I said, I've used Opera Mini for years, but those were on platforms where the stock browser was so bad or so slow that I put up with Mini's inconveniences. But on Android and specifically Evo, I could not recommend Dolphin more.