That would be the Barcelona-Madrid route, according to Travel Daily News, which cites data from OAG BACK Aviation Solutions. The data looks at the number of "flight operations" per week to determine the busiest routes. The Barcelona-Madrid route had 971 weekly operations, "closely followed by Sao Paulo Congonhas/Rio de Janeiro (894 per week), Jeju/Seoul Gimpo (858 per week) and Melbourne/Sydney (851 per week)," according to Travel Daily News.
Amsterdam-London Heathrow was the world's busiest international route with 350 operations per week. In North America, the busiest route is Honolulu-Kahului (639 weekly operations) followed by Las Vegas-Los Angeles (553) and San Diego-Los Angeles (514). Tied for fourth are routes from New York LaGuardia to both Boston and Washington Reagan National.
Of course, this list is based solely on frequencies, not actual passenger capacities; an airport pair served 400 times a week by a 44-seat Embraer Regional Jet still carries far fewer actual passengers than an airport pair served 200 times a week by a 240-seat Boeing 757. Also, the list is based on individual airports, not cities; the number of flights between two actual metropolitan areas might be larger due to the presence of multiple airports in each area (indeed, somebody commented on Ben's blog that the busiest domestic metro pair would actually be Los Angeles area and the Bay Area, with 840 weekly flights between the two).
Nevertheless, I've always wondered how Southwest's every-half-hour service between Houston Hobby and Dallas Love stacked up in relation to other heavily-traveled routes. According to Southwest's flight schedule, there are 372 commercial flights between the two airports every week.