But the main reason is that the British way simply makes more sense. Indeed, since at least the 1960s a common designation for that style has been "logical punctuation." The best way to grasp this is to look at an example, such as what Slate commenter Dean Hamer wrote under a recent article about PBS and NPR:

[I]ronically, given the anecdote about "Tales of the City", PBS is the ONLY widely available channel that has any serious LGBT content; e.g. documentaries such as "Ask Not" and "Out in the Silence".

"Tales of the City" and "Out in the Silence" are units—consisting of the words and the quotation marks. Insinuating a period or comma within the unit alters it in a rather underhanded manner. American style is inconsistent, moreover, because when it comes to other punctuation marks—semicolons, colons, exclamation points, question marks, dashes—we follow British/logical protocol. Dean Hamer would pass muster in any U.S. newspaper or magazine, for example, if he were to write: I am a big fan of "Tales of the City"; did anyone else see "Ask Not"?

I've always practiced the logical punctuation ever since it occurred to me that there's a difference between:

Have you seen "Apocalypse Now?"

and...

Have you seen "Apocalypse Now"?