From the Dairy Council website:
"Organic milk comes from cows that have been grazed on pasture that has no chemical fertilisers, pesticides or agrochemicals used on it.
The producers must register with an approved organic body and are subject to regular inspection.
Once the cows have been milked, the milk is treated in exactly the same way as regular pasteurised milk.
The nutritional content of organic milk is exactly the same as non-organic milk."
http://www.milk.co.uk/page.aspx?intPageID=43
There's actually some evidence which contradicts that last paragraph:
http://news.bbc.co.uk/1/hi/health/4153951.stm
but it seems unlikely that there are any specific health benefits from drinking organic milk. Similarly, the 'organic' label doesn't necessarily imply any higher standards of animal welfare.
Chris