The summer/winter/etc sale trading cards are similar to regular trading cards with a few exceptions, mainly the expiration date you mentioned, and that there isn't (usually) a level cap on the badges you can craft with them.

(If you were asking about trading cards in general, the badges you craft will let you gain levels. Each level increases your friend cap by 5 (up to 2000), and every 10th level gives you a nice showcase for your profile page.)

Gems can be sold on the community market, or be used to create booster packs, each of which gives you three trading cards for a specific game.

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