Desktop mode is for the people (vast majority) who cannot stand the "Metro" interface of Windows 8, myself included. When I bought my laptop, I wanted a computer. If I'd wanted a tablet of a phone, I would have bought a tablet or a phone. I never use the "Metro" tiled mode of WIndows 8 for anything. Frankly, it's confusing as all H E L L, and most of the apps don't work right, because my $279 laptop doesn't have the necessary hardware for Windows 8 Metro apps.
That's why there are two. So people can pretend their desktop or laptop is really a Windows phone. They're the same Internet Explorer, but with different graphic setttings. They use the same Flash Player plug-in (ActiveX) but for many, many people, Metro mode has all sorts of problems, like stuttering audio or choppy video, and full screen in YouTube works for some, doesn't for most.