"Titular" is a piece of quizbowlese used incorrectly to mean "of or pertaining to the title of something." In more common usage, "titular" is an English adjective meaning "in title only, as opposed to reality."
Following the Glorious Revolution, the descendants of James II were the titular monarchs of England.
Consider the following clue:
Binx Bolling is the titular character of Walker Percy's novel The Moviegoer.
In this sentence, "titular" modifies the word "character" to indicate that Binx Bolling is "The Moviegoer" of the novel. However, this requires awareness of this secondary meaning of the word; hence, the much more common and clear way to write this sentence would be:
Binx Bolling is the title character of Walker Percy's novel The Moviegoer.
This removes ambiguity about the intended meaning of the word and clarifies the relationship between the clue and the answer.
Neither word is appropriate to use when referring to "the titular event" or "the title phenomenon" in history or science questions that are not on formal and unambiguous published titles of creative works, as no titles in any sense are involved.
In quizbowl, the word "titular" used where "title" would have been much more appropriate became a meme around 1998, possibly due to HILARIOUS! evocation of a slang term for the female breasts. The intentional use of the word to provoke giggles (or, perhaps, tittering) subsided by 2003. Unfortunately, ever since 2010, a non-ironic use of "titular" has once again exploded as writers imitate the phrasing of past questions. In literally every case, it is better to use "title" as an adjective or consider whether multiple references to "the titular/title character" in your question are helpful in the first place.