According to the well-respected D&D historian Jon Peterson, the second-ever published RPG was neither Tunnels & Trolls nor Empire of the Petal Throne, but the fan game Rules to the Game of Dungeon by Craig VanGrasstek. Quoting John Peterson's Blog
One of the perennial questions about the history of role-playing games is this: which came second, Tunnels & Trolls or Empire of the Petal Throne? Deciding between the two is largely a question of semantics, of whether you count various small-run amateur publications as releases or not. Fortunately, historians don't need to choose between the two, because Craig VanGrasstek's Rules to the Game of Dungeon (1974) beat them both handily. Weighing in at eighteen pages, and released late in the summer of 1974, Rules to the Game of Dungeon seems certain to be the second published role-playing game.
It really is very difficult to answer the question of whether EPT or T&T were published first. Here is the chronology of events (mainly based on footnote 814 of Jon Peterson's excellent book Playing at the World, though also on the information given with the pdf release of the EPT orginal manuscript at Drive Thru RPG.)
- Prior to 1974: MAR Barker develops his Tekumel world.
- "Spring of 1974" Limited print run of 50 copies of EPT distributed to friends of MAR Barker.
- April 1974 Ken St Andre encounters D&D
- June 1975 Ken St Andre has 100 copies of T&T printed at his university print shop
- July 1 The Stragetic Review Announces EPT will be released "Mid July"
- July 3-5 1975 Ken St Andre sells 10 copies of T&T at Westercon
- July 25-27 TSR's print version of EPT released at Origins
So the question is - do you consider the private print run of "Spring 1974" to be publication? If so, EPT was third. If not, T&T was (probably). Note that we don't know exactly when TSR printed EPT.