There isn't a big time difference between the two routes if all goes well. They are probably so close that the area of Birmingham you will be departing from may have an effect on which is the better choice.
I like going the Troy route: I-65 to Montgomery, Hwy 231 to I-10, I-10 to I-75, south on I-75 to the Florida Turnpike, south on Florida Turnpike to Hwy 27, south on Hwy 27 to Hwy 192, east on Hwy 192.
Going through Atlanta is farther and when I have gone that way, it has taken a little longer. The traffic around Atlanta can be heavy as well as the construction and traffic on I-75 through Georgia. I've heard of too many people (and its happened to me) being "stuck" on I-75 due to an accident or construction.
Hwy 231 is mostly a divided highway with 65 mph speed limits. Troy and Dothan have traffic lights and turns which slow the drive down a bit.
After getting to Montgomery on I-65, I take I-85 North towards Atlanta (actually heading east while in Montgomery) for about 10 miles, then go south on Hwy 271. Hwy 271 South ends at Hwy 231; this bypasses traffic. When traveling east on I-85, Hwy 271 is the exit AFTER the exit for Hwy 231 (look at a map and you will see why this makes sense).
