Before we answer that question, let us see what the difference is between the bonnet hair dryer and the blow dryer. Basically speaking there is not much of a difference in their working at all. Both of them work by pumping in hot air on to your hair so that the evaporation process is sped up.
It has long been known that if you "set" your hair when it is damp and then dry it out, your hair will remain that way far longer than even if you use high tech creams to hold your hair in place. Plus you get the wavy soft look that creams cannot give you.
It is for this reason that the hand held hair dryers were invented, so that you can arrange your hair any way you want to without having to go to a hairdressers to do it. Most people use the blow dryer simply because it is faster, and the bonnet hair dryer is generally not preferred because it takes longer.
In actual fact the blow dryer is better only if you want to dry out your hair quickly. If you want to brush and set your hair in place, it is actually more difficult with a blow dryer. You will have to brush with one hand while blow drying your hair with the other, and most of us find this very difficult. This means that we take longer leading to heat damage to our hair. Then we have to go and spend a lot of money in finding something that will remove this damage, and the whole thing is just one big circle.
A bonnet hair dryer on the other hand is very simple. It has a unit that you mount on the wall and a cap that you put over your head. When you switch it on, hot air is blown into the cap heating your hair evenly. This gives you the option of you brushing your hair in place, putting on your cap and putting on your make-up while your hair dries.
Because the bonnet hair dryer heats your hair evenly, you can put it at a much lower temperature setting and dry your hair for longer. This will ensure that your hair does not get heated to such an extent that it gets damaged.
The only flip side to this is that you are tied down by the length of the hose that connects the cap with the mount. Many people also find it highly embarrassing to use one of these which is why they prefer to go with the blow dryer.
As hair dryers go the bonnet hair dryer is what is good for your hair. Of course it will not work as well with long hair, but with regards to preventing hair damage, it is the best.