Hey Eli...
SuperFlyer, that's not actually true. I assume you are referring to scan lines, and converter boxes. Converter boxes work fine for games and video, the only down side is that they upconvert/downconvert the pixels, so you end up with often odd-looking graphics up close, especially on text (which is what Eli was describing, it sounds like).
The fine line between televisions and computer monitors are blurring. Once upon a time, TV's used interlaced "scan lines" to draw pixels on a CRT screen, and you were limited to a resolution of roughly 480 horizontal lines. With the advent of Digital and High-Definition LCD TVs, they are starting to very closely resemble the same progressive scan technology that computer monitors use, and in fact often support higher resolution (a quick look at newegg.com's 22" monitor selection will show you that most PC monitors don't even support 1080p resolution!).
The result is that new TV's are often more expensive than PC Monitors, when it used to be the other way around. In many cases, you could use either for both, its really a question of what connections are available. I've seen TV's with VGA ports, and computer monitors with HDMI (or even older analog RCA ports).
I admit I could be wrong about some of this, but here's how it works as far as I understand it:
The big difference between digital TVs and Computer Monitors nowadays is mostly the adaptable resolution and that TVs still have support for older composite connections (the analog RCA tri-colored, for example, or 75Ohm cable wire) and built-in speakers.
HDMI is being adopted as a new standard for video components, however it is still mostly used for TVs because it includes audio in the connection, whereas PC's don't need usually handle audio in the monitor. Also, PCs can handle the same video in older DVI or even VGA connections, so there isn't as much of a reason to adopt a new standard.
Other than that, they seem to be mostly interchangeable. It sounds like the problem you are describing has more to do with resolution inconsistencies that monitor types. While CRTs used to be able to adapt to different resolutions, LCD screens have a set amount of pixels no matter what. The end result is that if you turn DOWN the resolution, you'll start to see pixels doubling up on each other to fit the screen. If you take your PC with a 1440x900 resolution monitor, and turn it down to 640x480, you'll see significant "pixelization" like you describe.
If you are using an HDMI screen, you must make sure your PC is outputting in exactly HDMI resolution (720 or 1080) to get a sharp picture.