I've camped both places and you need the same basic items--tent, sleeping bags (it gets cold in the desert), food, cooking equipment. The difference is that you need a lot more water in the desert. There's probably a bigger temperature differential between day and night in the desert than in the woods.
No need for different equipment. Just bring more water in desert situations. Be prepared for cold conditions, rain, wind, hot weather, drenching sun and high winds wherever you go. Camping is a great experience that is relatively inexpensive once you get your equipment. Buy a good tent to house yourself.
Well, yes. You can leave the water purifier at home and stock up on gallon jugs of water which you would not lug into the woods. You will also desperately need head protection in the desert and with no firewood available, you must have something else to cook with and frighten snakes; cactus is difficult to burn.
I have also experienced high winds in the dessert. You have a feeling the whole tent will fly away. You need to make sure the stakes are positioned firmly in the ground. Usually this is not an issue in the woods. Also load up on suntan lotion.
Yes. These two ecosystems are significantly different and present different challenges to camping. If you were camping in the desert you would want your tent to be optimized to stay cool on the inside. If you were camping in the woods, you would need a tent optimized to stay dry in case of rain.