Yes. In some languages those are actually not "markings" but denote proper letters, like in German ä,ö,ü and ß. But even if not, like in French, it can alter the meaning of words. E.g la != là. Therefore, for most Europeans and speakers of other languages that depend on more letters than ASCII provides, it is very annoying when that is not supported properly.
However, I have made the experience in a few cases that particularly Americans have a hard time understanding this. The remark about your wife not caring seems to be in this vein, too. Recently, I decided to convert our MySQL DB tables from latin1 to UTF8. (I wasn't even aware that we didn't have some form unicode, as our DB is only few years old, and I thought some unicode is the default nowadays everywhere. But then MySQL...)
Anyway, my CEO (also an American incidentally) was trying to keep me from it because he thought it's not high priority. However, we're about to go live in a French-speaking region, but which also has other indigenous languages (and therefore names), with their own "special" characters (I put "special" in quotes because for those languages, they're not "special" at all -- but I guess you get my gist by now).
Also, in previous jobs I have converted legacy systems to unicode and know what a pain it is down the road. Not to mention all the hard-to-find bugs if you don't do it, because some strings don't compare as they should, or people are just annoyed because their name is not shown correctly.
So I went ahead with the conversion anyway. We may never know for sure, but I'm convinced that I saved us some major customer frustrations, days of bug hunting and weeks of converting everything later, when existing data would need to be migrated.
So please everyone, just use UTF8 or some other unicode variant from the get-go. The few bits you might save otherwise are just not worth it.