A "card" is a flat rectangle. It is often carried in a wallet. A credit card, a library card, a business card, and an ID card are physically the same, but are for different purposes. In the United States cards are often a standard size, 3.375 inches wide by 2.125 inches high (86.6 x 54 mm).
If someone wants to see your ID card, you take out your wallet, take the card out of the wallet, and hand them the card.
A "badge" is something that you wear on you, often on the front of a shirt or business jacket. It's always out and in view. If someone wants to see your ID badge, all they have to do is look at you. An "ID badge" is often a printed card, often with a photo, in a plastic holder. It can be pinned or clipped to a shirt, or hung from a belt.