I think the reason is that your path for the background image you use in your menu on your index page isn't the right one.
Eg, when I look at the path of that image on your about page, it reads: http://blog.moltenwords.net/wp-images/bg1.gif
And that image is there when I type that in the url bar. Contrary, when I look at the path of that image on your index page, it says to look for the image at the following location: http://blog.moltenwords.net/wp-content/themes/Carrots/wp-images/bg1.gif
And when I look there, there's no image.
This is because you copied the stylesheet verbatim. But the relative location of that image didn't stay the same.
Conclusion: the image in your index.php page can't be found so it displays none. The solution is to simply change the url (relative or absolute, your choice) to the correct location of that image.