The major difference is in the order of the numbers when we only use numbers for the date. Barb's example above (4/15/2012) reads April 15th 2012 to her. To me, it's an impossible date because the middle number should be the month and there are not 15 months.
April 15th 2012 for me = 15/4/2012. It doesn't matter whether I say "April the fifteenth" or "The fifteenth of April", it's always 15/4.
As far as I know, the only date which is regularly used in the American construction in the UK (and I imagine in most of the world) is 9/11, just because that is how we have become used to hearing and seeing it for over ten years. However, if you showed that date written down to someone in the UK who had never heard of the World Trade Center disaster, they would tell you that something happened on the 9th of November.