OK, so I guess moshu was just tweaking me and not actually offering any help. Here's what I've discovered about the Google Maps html code: If you accept the driving directions that Google Maps gives you, the code will work perfectly, and you'll have a nice map embedded in your page.
However, if you try to modify the route by dragging the lines in Google Maps, the code doesn't work at all, and the route is completely messed up. And even if you change the default route by adding intermediate destinations (rather than dragging the lines), the route is still messed up.
Has anyone encountered this and found a work-around?