Catch and chase are the same word, borrowed twice. Both go back to Vulgar Latin captiare, "to try to seize," from Latin capere, "to take." The difference is which French dialect they came through. Catch came about 1200 from Anglo-French or Old North French cachier; chase came a century later from central Old French chacier. Norman French kept a hard k where Parisian French had softened it to ch, so English ended up with both.
That k-versus-ch split is one of the most useful things to know about English vocabulary, because it happened repeatedly: warden and guardian, warranty and guarantee, cattle and chattel are the same pairs of Norman and Parisian forms. In early Middle English catch still meant "to hunt," a sense it later handed over to chase. The -tch spelling, meanwhile, is native English convention, applied here to a French import.
catch (v.) c. 1200, "to take, capture," from Anglo-French or Old North French cachier "catch, capture" animals (Old French chacier "hunt, pursue, drive" animals, Modern French chasser "to hunt"), from Vulgar Latin *captiare "try to seize, chase" (also source of Spanish cazar , Italian cacciare ), from Latin captare "to take, hold," frequentative of capere "to take, hold" (from PIE root *kap- "to grasp"). A doublet of chase (v.). Its senses in early Middle English also included "to chase, hunt," which later went with chase (v.). Of sleep, etc., from early 14c.; of infections from 1540s; of fire from 1734 (compare Greek aptō "fasten, join, attach, grasp, touch," also "light, kindle, set on fire, catch on fire"). Related: Catched (obsolete); catching ; caught . The meaning "act as a catcher in baseball" is recorded from 1865. To catch on "apprehend, understand" is by 1884, American English colloquial. To catch the eye "draw the attention" is attested by 1718. Catch as catch can has roots in late 14c. ( cacche who that cacche might ). also from c. 1200 catch (n.) late 14c., "device to hold a latch of a door," also "a trap;" also "a fishing vessel," from catch (v.). The meaning "action of catching" is attested from 1570s. The meaning "that which is caught or worth catching" (later especially of spouses) is from 1590s. The sense of "hidden cost, qualification, etc.; something by which the unwary may be entrapped" is slang first attested 1855 in writings of P.T. Barnum. also from late 14c.