There is no difference between catsup, ketchup, and catchup; those words are all commonly used to describe the tomato-based condiment. Technically, the words can apply to any sauce made from “the juice of mushrooms, walnuts, tomatoes, etc., and used as a condiment,” and they are usually qualified with the main ingredient (e.g., Heinz Tomato Ketchup), says the Oxford English Dictionary.
The New Food Lover’s Companion notes that the origin of the words (and the sauce) is ke-tsiap, a fish-based condiment that was popular in 17th-century China. According to Tracey Parsons, a spokesperson for Heinz, British sailors discovered ke-tsiap in the 18th century and brought it back to England, where people began tinkering with it. The early recipes published in 18th-century Great Britain called for “kidney beans, mushrooms, anchovies, and walnuts,” writes Andrew F. Smith in Pure Ketchup. Then the condiment spread to the American Colonies, where, Food Lover’s notes, tomatoes were added in the late 1700s.
The OED cites catchup as the oldest of the three variations, with the first citation appearing in 1690. Ketchup comes next, in 1711, and finally catsup appears in 1730. Parsons says that Heinz originally went with catsup, but the spelling was changed to ketchup in the late 1880s as a way to stand out from the competition, which was using the catsup variation. Now ketchup is the most commonly accepted spelling in the industry.