The most common example is probably that i (U+0069 LATIN SMALL LETTER I) and I (U+0049 LATIN CAPITAL LETTER I) transform into each other in most locales, but not all. In locales az and tr (the Turkic languages), i uppercases to İ (U+0130 LATIN CAPITAL LETTER I WITH DOT ABOVE), and I lowercases to ı (U+0131 LATIN SMALL LETTER DOTLESS I).
case-insensitive is all fine if you only handle text that consist of A-Za-z, but as soon as you want to write software that works for all languages it becomes a mess.