Ketchup - the word "ketchup" originated around the year 1711 to describe a fish sauce called "kecap" from either China or Malaysia. Scholars can't seem to agree on which, so it is open for debate. Sometimes it was written as "catchup." The tomato-based sauce that we now call "ketchup" arrived in the early 1800's in recipes, but there was also a mushroom ketchup at this time. (Mushroom ketchup is still sold in Britain today, and banana ketchup is sold in the Philippines.) When the Heinz company started selling bottled "tomato ketchup" in 1876, their ketchup became the most popular tomato-based ketchup sold in the US, so the "tomato" was dropped and the word "ketchup" became synonymous with the thick tomato condiment we know and love today, similar to the way that most people say "hand me a kleenex" instead of "hand me a facial tissue," no matter what brand of tissue it is.
Catsup - the word "catsup" first showed up in Poems Composed at Market Hill by Jonathan Swift in 1730 when he wrote about "Botargo, catsup, and caviare." For a while the word "catsup" was more commonly used in North America, and then "ketchup" took over due to large-scale manufacturers like Heinz and Hunt's calling their product "ketchup."
Short answer: ketchup and catsup are the same thing; a tomato-based condiment with vinegar and spices.