Magical - aside from WW, the JL are at a disadvantage here in handling the supernatural. The Doctor should be able to handle pretty much any magical threats all by his lonesome.
Scientific - Batman has better prep feats than Engineer, and Brainiac is one of Superman's primary villains who he's successfully faced off against countless times, so maybe a slight edge to the JL in resolving a tech threat.
Alien invasions - This is JL all day.
Natural disasters and saving civilians - I wanna say JL again. But Doctor has shown he could protect and wall off large sections of a city to save civilians. The Authority have the benefit of doors to travel around different locations instantaneously to save people from disasters, but JL has Flash...
Cosmic events - Toss up, with maybe an edge to JL again. I'd say JL has more feats of being successful against such threats, but I don't know if they were more efficient at resolving them than the Authority. Authority's threats like "God" returning to Earth and the Renegade Doctor seemed less threatening than some of the things the JL had to face off against, like the Crises, Mageddon, etc.
The old Govt turns against them gambit - definitely The Authority. They tried to take down the US Gov't and even succeeded in taking over the country for a while... the JL seem more likely to try and negotiate or even placate the gov't out of their respect for the law, where as The Authority wouldn't blink twice at killing off any problematic gov't officials...