We are often asked for a bid on screen printing hats. Unfortunately, on most headwear this is not possible. There are some work-a-rounds, but for the most part hats have to be embroidered.
Many hats have seams right down the center. This would prevent the screen from making good contact on the hat. In addition the fabric used in hat construction tends to be very textured itself. The bill would be in the way when trying to print. As well as the inability to get the hat to lay flat while passing through the dryer (it's like a pizza oven). In short it cannot be done.
Enter the trucker hat. Trucker Hats do not have the seam down the center. They have much less texture on the printing surface. They are usually lighter, and not as durable as their twill counter parts. However they are extremely popular, and cheaper than other hats.
The trucker hat still cannot be screen printed. I think they would melt in the dryer -- I have never tried it. By using a heat press or dye sublimation with a heat press, a similar effect can be reached.
The most popular technique is still embroidery. If done correctly the end-product is very impressive, and durable.
How do you choose between the two? Ask yourself who will be wearing these? If you are producing hats that are going to be sold at a rock concert, I would recommend the trucker hat. If you are looking to give these hats away to your employees, embroidery is the answer.