It's a very good question.
I feel that the previous two answers are combing 'organic' with 'free range' and 'non-organic' with 'factory'.
Hormones are not allowed to be used as therapeutic agents or in feed in UK farm animals. Antimicrobial growth promoters have also been banned in the EU, so farm animals should not be eating antibiotics either!!!
Animals in non-organic systems do not get antibiotics 'routinely in case they get ill', they get them if they get a specified bacterial condition which the antibiotics would improve.
In answer to the original question, animals in organic systems get the same veterinary treatment as animals in non-organic systems. However the withdrawal periods for these drugs (e.g. antibiotics) are longer than with standard systems. Where organic systems differ is that prophylactic treatments e.g. vaccines are less lilely to be used. They may still be used when there is evidence of a particular disease. Organophosphates are not allowed to be used at all.
In organic systems, groups of animals aren't allowed to be treated (for instance in a bacterial disease outbreak, you can't treat all the animals in contact with the affected animal with antibiotics) but rather other management factors are used to try and reduce disease spread.
It is true that in some places (and I have experienced this myself) that the animals are less likely to get veterinary attention when they are sick, and this can lead to decreased animal welfare. In some organic farms we do see more disease, and that can be due to reduced likelihood to use certain drugs.
However in most cases farmers are quick to get disease and illness under control.
The Soil Association have an article here:
15:56 Tue 26th Feb 2008