Playing cards are NOT tarot cards. Tarot cards are a late offshoot of playing cards.
The oldest playing cards were used in ancient china, and the 4-suit 52-card deck we used today was in use by the Mamluks in the 1300s (it is unknown when the 13th card was added to each suit - 48-card decks with 4 suits were in use in the 1000s and seem to have inspired both the 8-suits-of-12 decks of India and the 4-suits-of-13 Mamluk decks), using polo-sticks, swords, coins, and cups. When cards caught on in Europe, the Polo-Stick became the cudgel (as polo was an almost-unknown game in Europe), and in the mid-1500s a new five suit deck was created, probably inspired by the 48-card (some European nations omitted the ace for a couple of centuries) game Karnöffel that featured a complicated predecessor to the trumps used in games like euchre. In this new deck, the fifth suit would simplify the game by always counting as trump. This 5-suit deck was more expensive and thus remained the exclusive plaything of the upper class, with the lower classes sticking to the older one.
As the deck passed through Germanic countries, the suits became Leaves, Hearts, Bells, and Acorns before gestation in France morphed these into Clovers, Hearts, Pikes, and Tiles. When these moved to England, a fair smattering of the old names was still in use, and the mixture resulted in the Clover being called the club, and the Pike became the Spada (sword) and eventually Spade, as the older name was applied to them.
Tarot, with the odd fifth suit, languished in upper-class obscurity for about two hundred and fifty years until the late 1700s, when one of the perennial tides of mysticism siezed on them as a fortune-telling device. The Trump suite became the major arcana, and the other four suits constitute the minor arcana. Clubs became wands because that's a more mystical name, and for a long time it was common to call the coins "pentacles" or "circles" (both elements of traditional magic). The Wand is thus the equivalent of the club, and the Sword is a spade.