The difference between deodorant and antiperspirant
The simple explanation is deodorant stops the smell and antiperspirant stops the sweat, but there is more to it that your armpits stink because of apocrine sweat glands. These glands are all over your skin, but they are highly concentrated on your underarm. Apocrine glands excretes fats and proteins which feeds bacteria and creates stinky smell.
Deodorants work by targeting that bacteria. Many deodorants contain a chemical called triclosan. It makes your skin too salty or acidic to suppress bacteria. It won't keep you dry, but keeping the bacterial under control you won't stink.
Antiperspirants stop you from sweating in the first place. Ingredients like Aluminum and zircronium plug sweat glands. Any bacteria on your armpits won't have anything to eat. You can also avoid smelling bad by changing your diet. Certain foods like broccoli and cabbage can make body odor worse.
SHOWERING HELPS TOO !