I don't know if this is the only difference, but from what I understand, the kosher turkeys have been brined; meaning soaked in a salt-water solution, which I know doesn't sound good but it is supposed to make it very moist and flavorful.
I tried brining a turkey one year, but the brine I used was a recipe I'd found which consisted of apple cider, salt, and a bunch of spices, and you had to boil it for a certain amount of time and then let it cool, then soak the turkey in it for (I think) a whole day, then take it out and let it sit in the refrigerator for a day, all before cooking. I have to admit, the turkey was delicious, but that's just way too much work if you ask me. Plus, you have to have a big enough container and room in your refrigerator, neither of which I usually have. The year I did it I ended up using a plastic dishpan, but it was filled to the very top and difficult to move around, and the turkey had to be turned every few hours.
Oh - and I know you're thinking, "Wouldn't it make it very salty?" - you would think so, right? But apparently for some reason letting it sit outside the brine for the last day somehow mitigates the saltiness.
Anyway, the instructions I found on line with the brine recipe said that an alternative to brining the turkey yourself would be to get a kosher one. I've never been able to find one, though!