You are correct mostly. It has to do with the resolution differences.
Pictures on web sites are usually screenshots taken of the game in its native resolution. They are usually very low res, below even 640x480. In Windows when you increase the resolution of your desktop, you gain more space because it is displaying more pixels per dot than in a lower resolution. As you increase the number of pixels displayed the image will get smaller because there is more space on the screen, but the sharpness and clarity will increase to a degree.
This is why modern games look very good and crisp in higher resolutions. Although they do also have high resolution graphics (to keep the size from being too small as you increase the resolution) too.
With dos games running in full screen on a modern monitor, 17" or larger, and especially for LCD's, you will get more pixelation because they are higher resolution displays. Although if you're lucky, with an LCD you may get some blurrying from the non-standard resolution (all LCD's are optimized for a particular resolution, called the native resolution) and that blurrying might help the picture look better.
Unlike DOS games, emulators actually overcome this problem because they feature graphics filters that can enhance the image through smoothing and sharpening / interpolating techniques.. I don't know of any DOS emulator/environment that can do that yet, though I think DOSBox MIGHT have some options, I can't confirm it as I haven't used it in forever.
Most DOS and older games were made back when we had smaller monitors like 15" or smaller, and as such the picture looked a little better too. I used to have an old Apple monitor I got from school, with a composite input (yellow RCA) that I would use for my Playstation. Compared to televisions which had larger screens, my picture was always much better looking and sharper/detailed because of the smaller screen size.. It also had excellent color balance too.