Documented: both forms descend from the same Chinese word. Cha spread overland through Central Asia, Persia (chay), Russia (chai), India, and the Arab world. Te, the Minnan (Hokkien) pronunciation, spread by sea with Dutch traders from Amoy ports in the seventeenth century, giving Dutch thee, French thé, English tea. Polish herbata and Portuguese chá are the famous exceptions, Portugal traded through Cantonese Macao, not Amoy.
The observation linguists love: an entire planetary vocabulary hangs on one regional pronunciation difference, whether the initial consonant is ch or t. A dialect feature of coastal Fujian became the default word for half the species because Fujian owned the sea lanes.
The deeper reading: cha cultures built the tea ceremony inward, meditation and measure, while te cultures built the tea trade outward, auctions, clippers, and the Opium Wars. The word you use predicts which history your teacup carries.
What does your word for tea say about the ships your ancestors watched?