Typically with PDF you use vector fonts: the font designer creates the shapes of the letters, which are encoded as outlines. Your PDF viewer takes these outlines, and given your chosen zoom level and monitor resolution (which together determine how many pixels are available for displaying the letter "e", say), it rasterizes (converts to pixels) on-the-fly. In doing this (for low-resolution devices like computer monitors) it is further aided by “font hinting” which specifies how to hammer the font shape into the available pixel grid.

On the other hand, given Knuth's goals of not having to re-do all the work when the technology changes, METAFONT is an “end-to-end” solution: starting with the font design (at a higher level than the outlines that are ultimately shipped for vector fonts), it goes all the way to generate the actual pixels for a given resolution (like 600 dpi or 2400 dpi), resulting in bitmap fonts (pre-rasterized) rather than vector fonts (rasterized on-the-fly by PDF viewer).

There are a few consequences of these differences, for on-screen viewing:

• Computer monitors are very low-resolution devices (even the most high-end “retina display”, “4K” or whatever) compared to print. There are several tricks required to make text look good on screen (see e.g. the chapters at rastertragedy.com for details): things like anti-aliasing/grayscaling (using pixels that are neither black nor white) and subpixel-rendering (using individual RGB subpixels that make up each screen pixel).

• There's a conflict between thinking of on-screen viewing as serving to approximate to what the printed page will look like, versus allowing deformations or even re-positioning characters from where they'd be printed, to simply look better on screen.

But all of this has nothing to do with the DVI format, ultimately (which is just an efficient encoding of the page layout, i.e. what characters from what fonts go where on the page). It's just that (1) many DVI viewers tend to use bitmap fonts for historical reasons, resulting in poor on-screen viewing, (2) some DVI viewers use their own libraries for rendering instead of using the best font-rendering libraries available on the system.

• Even if using a bitmap font, the DVI viewer has a choice: (1) it can take the bitmaps that were generated for a certain resolution (typically for print, so something like 600 dpi which is probably higher than your screen resolution), and try to render those bitmaps at the number of pixels available (with the typical problems that result), or (2) it can try to ask METAFONT to run again to generate the fonts at whatever low resolution is appropriate for your monitor (but still without anti-aliasing and subpixel rendering, as MF was designed for higher-resolution devices and doesn't bother with all that). DVI viewers seem to always take the first option, probably to prevent the user's disk from filling up with lots of bitmap fonts for every possible zoom level. So the font rendering ends up sub-par. If you print it on a high-resolution device the output will be fine.

• For screen display, the DVI viewer could choose to look up and use the corresponding vector font.

• Finally, the DVI viewer could just convert to PDF (with dvipdfm or whatever) and have the same on-screen experience as with something that was originally a PDF file. In fact, on my current macOS system, the two DVI-viewing options I have (Skim and TeXShop's viewer) both seem to do exactly that internally, and the output is excellent.