Surprisingly, not many! The biggest downfall is the fact that it is not instantly ready to ride and require inflation. While this will only set you back about ten minutes, if you forget the pump, you'll be out of luck for the whole expedition. You will also need to set a couple of minutes aside to attach accessories, such as the leash and fins.
Another common complaint about inflatable paddle boards is their style. They aren't quite as aesthetically pleasing as regular paddle boards. It does seem as though companies have taken notice of this and have been improving the designs.
Overall, for us, we feel as though the good outweighs the bad here and in the end, it is a personal decision. Either choice you make, you're making a great investment for your health and happiness.