A quick search through Newspapers.com for "Bat Materson" between 1874 and 1924 (the year after Bat's death} approxmately 16,000 articles. Of course, a lot of these are duplicates or close matches that are not actually Bat. For "Wyatt Earp" the number is close to 4,500, of course, with the same caveats. Because his name was misspelled so often, a few more articles might be found using "Erp", Earpp" or even "Erpps", but much less than a thousand.
So I would say that over the same time they were on earth, Bat was better known to the general public, but largely because of his long term connection to boxing and the fact that he had a weekly sports column for the last decade or so of his life. At various times between 1874 and 1900, they would exchange places in the national spotlight (such as coverage of the street fight and vendetta ride for Wyatt and the Battle of the Plazza for Bat) and sometimes share it (as in the Dodge City War). Sometimes they were virtually each others publicity agents, such as Wyatt's article about Bat in the 1896 Examiner aricles and Bat's article on Wyatt in HUMAN LIFE in the early 1900's.