Computer modern is the default font for TeX because it was created at (more or less) the same time as TeX by the same author, specifically for that purpose. For some time it was essentially the only font set practically usable with the TeX system.
The defaults in LaTeX or plain TeX never change as you should be able to process a document from last century and get the same page breaks.
The glyph makes much more sense visually when seen as a ligature of long s and round s, one of the two traditional forms of the ß (the other, of course, being long s and z). Here's a comparison, using outlines from cm-unicode, version 0.6.3a:
Here I've used f as a reference for the first part of the ligature, since I couldn't find a long s in cm-unicode. So the "super-thin" line you mention in your comment is thin because it's only acting as a ligature between the two sub-glyphs. You can see that the s part has been compressed horizontally to keep the total width reasonable, but on the vertical axis it matches up very closely.
Addendum: having discovered from other answers that Jörg Knappen was the designer of this ß, I was able to find a few contemporary Usenet postings relating to the design. The most concise explanation seems to be this one, from Knappen himself:
I consciously redesigned the sharp s to exhibit the
ligature structure <long s> <short s>. Despite the
popolar name eszett I find the arguments in favour
of this analysis (as given by Tschiold) more convincing
than the ones in favour of <long> <z>.
There's also a thread in German which can be summarized as a vigorous discussion over the relative merits of the ſs and ſʒ forms, in which Knappen firmly defends his choice.
One remaining question is why Knappen made the upper part so narrow, when many (most?) fonts with the ſs variant bring the ligature line out much further to the right -- Linotype Aldus Roman, for example:
In this case I don't think there's anything on record, and we have to assume it's a personal stylistic choice. This "compressed" style of ſs certainly isn't unique to Computer Modern. For example, here it is in Antiqua:
As to actually replacing this glyph with something you find more appealing: if your TeX installation is sufficiently recent, switching to Latin Modern should be enough. From Martin Schröder's answer and barbara beeton's comments, it seems that the Knappen ß was the default in some older releases of Latin Modern. So if a simple \usepackage{lmodern} doesn't do the trick, you could consider updating your TeX installation, or following Ulrike Fischer's instructions for selecting the Knuth ß from the cm-super fonts.
Best Answer
If you make a LaTeX document without using many packages and without altering other defaults, then the style of the document is largely determined by two factors: the default font designed by Donald Knuth, and the default layout of the standard document classes designed by Leslie Lamport.
The Font
The default font is Computer Modern, designed by Donald Knuth. It is a computer version of the "modern" style typeface that was used in the first editions of his book The Art of Computer Programming. Knuth created TeX with the primary goal of using the computer to typeset this book.
Page Layout, Margins, Headings, etc.
The basic LaTeX document classes were designed by Leslie Lamport, based on then-common conventions for scientific publishing. For example, the large default margins in the
articleclass are intended for use by academic journals, which have a narrow type block, and would trim down the page to fit their format.These core classes have most of the style elements hard-coded into them, meaning that the style decisions cannot easily be modified by the user. This was because of the memory restrictions of computers at the time. Now we can use packages to alter these values, or we can use alternate classes that make it easier to customize.
Paragraph Shape, Character Spacing, Kerning, etc.
A third factor that shapes the overall look of any LaTeX document is the unsurpassed algorithm for building lines, paragraphs, and pages, used in the core TeX typesetting program written by Donald Knuth. This central feature of the TeX program controls the distribution of words on the page, which creates what designers call the "grayness" of the page. Some people claim to be able to recognize TeX-produced documents based on this alone.
Going Beyond the Defaults
If you use a different font package (try
ebgaramondornewpxtextorgilliusfor example) the look will change dramatically. If you use a non-standard document class (beamer,memoir, the KOMA-script classes) or customize the standard ones by using packages likegeometry, or write your own class, you can have any page layout and style you want.You can also use an alternate format, ConTeXt, which provides its own interface for customizing the layout.
Or you can use Knuth's original Plain TeX format, though this requires you to write almost all the formatting code yourself, down to exactly how much stretchable space to insert after a section heading or in front of a bullet point.
Donald Knuth typeset the later editions of The Art of Computer Programming using his own macro package for Plain TeX, and if you look at that book you'll see his stylistic preferences are rather different from those of the default LaTeX classes, such as using sans-serif font for headings and slanted type for book titles.