For years, rumours have abounded that there is a link between cancer and aluminium and parabens found in antiperspirants. Although that may not be proven, what is certain is that many chemical antiperspirants are made by companies which test on animals.
Antiperspirants minimise sweating by blocking the sweat glands with aluminium salts, while deodorants allow you to sweat, simply masking any odour. And it isn't actually the sweat that smells, BO comes from the bacteria in our skin breaking down the acid in perspiration. The best natural deodorants therefore will have anti-bacterial ingredients to keep any unwanted smells at bay.
Bear in mind that you will notice a difference when you make the switch from a chemical antiperspirant to a natural deodorant as your sweat glands which have been clogged by aluminium salts will have... well, a bit more freedom to perform their proper bodily functions. Though the floodgates won't open, there may be a transitional stage when you're more aware of your underarm sweating before your body reaches its natural equilibrium.