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A padded bra is a broad category of bras that includes any bra that has padding. Padding, however, has a broad definition. A lightly lined bra with two millimeter thick foam throughout the cup falls into the padded bra category, as does a push-up bra that has so much padding, it adds two cup sizes.
Not all padded bras are push-up bras, but all push-up bras fall under the padded bra category.