As you mentioned, Windows is based on DOS technology. Mac on the other hand is based off Unix. This is one of the reasons it is so stable, it runs off the same build blocks as Linux and other *nix systems.
Until recently, there was a marked difference between the functions of the two operating systems. For example, take Windows XP and Mac OS X. Windows XP and previous builds were built for productivity. Plain and simple they are workhorses. You can churn out documents all day long. You can hold masses of data and databases, and it all runs very well. However, while XP could run multimedia, it doesn't come close to what OS X can do.
OS X is built to showcase multimedia. This makes it very good at graphics intensive operations. That's why professional image and video workers use OS X. It's just better at creating multimedia. Not only that, but if set up correctly, it shows true colors better. Granted, you need a thousand dollar monitor to do things properly, but OS X is built to create videos, pictures and all kinds of multimedia.
Nowadays the lines have started to blur a bit. While the two operating systems are still very good at what they were intended for, they have started to move into each others territory a bit more. Windows is becoming better and better at multimedia and OS X is becoming better and better at handling productivity tasks that Windows excels at.
There are other differences as well, other than the core architecture and function. Macs are extremely aestically pleasing. They're built to be beautiful. It works well with the graphics functionality. PCs might be a bit less appealing to eyes, however they're much more appealing in terms of customization. You can customize so much more of Windows and PCs than you can Macs. Not only can the hardware be stripped apart and put back together with cannabilized parts, but the operating system can function much the same as well. Macs on the other hand can be customized to a certain extent, but in order to build a mac from the ground up you need a lot more knowhow and access to specific parts and codes. It's exceedingly hard to customize macs because installing drivers is harder to do than it is with PCs. While a graphics company can come up with a new graphics card and just throw it in a Windows machine, it doesn't work so well on mac. If the drivers aren't there, the operating system may not even start.
Macs are extremely expensive. Prohibitively expensive. Why? I don't really know to tell you the truth. You could say that the hardware itself is expensive, which it is, but the same hardware on the PC market can sometimes be half as expensive as the Mac side. Then again, everything on the mac side is refined down to the smallest level. Look at the computers, they look less like computers than artwork.
So to answer your main question about how they run, Windows is based on the DOS architecture and Mac on Unix. That means that they are suited to specific things. One thing I find a bit funny is that everyone says Macs are unique and PCs are mass produced. If you think about it, the core architecture of a PC is only used by PCs. Meanwhile the core architecture of a Mac is used by anything with a *nix.
I have a Macbook Pro and have used Macs for about 8 years now. I love them to death and they work wonderfully. However, I have to say that I love PCs just as much. All for different reasons. If I was going to use a computer just to watch movies and leisurely surf the web I would prefer a Mac all the way. However, if attempting to have fun customizing and getting work done, I'll go with a PC. Thank god that Apple switched to Intel processors and I can dual boot between OS X and W7