In addition to the reasons listed by @ShreevatsaR why the T1 font encoding is advisable even when writing (primarily) in the English language, there are two more reasons that were missing from his list:
TeX is only able to apply ligatures and kernings between characters when these characters are real glyphs from the same font. In OT1 (with 128 glyphs) you only have more or less ASCII characters and all diacritics etc are missing.
Searching in the output is not working whenever a diacritic character is being used as that ends up being a complicated box structure in the output and not a character.
So to stay with your example of the occasional
G\"odel: if the font designer has decided that because of the shape of the
G it needs some kerning to a following
ö then he can specify this in T1 but not in OT1 (for the
ö as that is not a single glyph in that font encoding). And there are a lot of kerning adjustments between characters.
The second point means that if somebody is searching through your papers (put up on the web as pdf's, for example) for the name
Gödel, then the name wouldn't be found.
So in short the T1 fonts simply give slightly better output whenever there is a single diacritic char used, because kerning is better, hyphenation still works, cut-and-paste still works from the output and searching in the output works properly as well.