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As a native English speaker, I've mostly been allowed the luxury of pretending that ASCII is enough, and have been able to treat font encodings as not my problem. I've seen lots of advice that I ought to include \usepackage[T1]{fontenc} in my preamble. (See, for instance, p. 337 of The LaTeX Companion.) What isn't really adequately explained is why I ought to do so.

So, if I am writing in English and have to make only occasional use of things like G\"odel, what does following the advice really get me?

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Note that this is font encoding (determines what kind of font is used), not input encoding.

The default font encoding (OT1) of TeX is 7-bit and uses fonts that have 128 glyphs, and so do not include the accented characters as individual glyphs. So a letter ö is made by adding an accent to the existing 'o' glyph.

The T1 font encoding is an 8-bit encoding and uses fonts that have 256 glyphs. So an 'ö' is an actual single glyph in the font. Many of the older fonts have had T1 variants devised for them as well, and many newer fonts are available only in T1. I think "Computer Modern" was originally OT1, and "Latin Modern" is T1. (Look at OT1 font encoding and T1 font encoding.)

If you don't use \usepackage[T1]{fontenc},

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Yes, hyphenation for languages with accented characters is the main reason that requires T1 font encoding. – Jukka Suomela Jul 30 '10 at 18:58
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Yep, definitely load lmodern (or some other font package) when you use T1. – Will Robertson Jul 31 '10 at 2:37
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The bigger problem is that without T1 you cannot copy/paste a name with a non-ascii glyph without getting them split up into their components. – Christopher Oezbek Aug 13 '10 at 16:43
@Christopher: You will have the same problem even if you use T1 in combination with something like ae. – Jukka Suomela Aug 15 '10 at 19:18
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@Jukka: I am not sure why you would still use the ae package. – Christopher Oezbek Aug 22 '10 at 10:04
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I think everybody should use utf8. Look at things like G\"odel you mentioned -- with such use, you wouldn't be able to copy it properly from pdf; with utf8 you'd just type Gödel in source, and it would just work.

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For me, at least, remembering how to type G\"odel is easier than remembering how to type Gödel. (Indeed, I had to C&P from your answer to avoid a google for how to type it directly.) – vanden Jul 30 '10 at 18:19
There are tools both on Windows and Linux which provide tables of all glyphs; also sometimes all you have to do is to copy from a webpage (as in this case). – mbq Aug 1 '10 at 0:48
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@vanden: to each his own; Mac OS X makes it very easy to type most accented Latin letters. – Will Robertson Aug 3 '10 at 6:19
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You're talking about the input encoding (and I agree, utf8 is the way to go there) but the question is about font encodings. – Zack Aug 7 '10 at 23:27
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I'm not sure that even makes sense; unicode isn't a list of glyphs, and there are strange things in fonts (especially math fonts) that are not even really glyphs. – Zack Aug 15 '10 at 19:11
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