I second the advice to study the Alabama route before going; though I don't necessarily recommend it at all times. I think that the Chattanooga route can be fine if you do it at the right time of day and in the right weather, but like the PP, I find that the Alabama route eliminates those worries. It *is* longer, though, so I reserve it as a backup choice.
Two factors make the Chattanooga route problematic: one is Atlanta traffic, which is relatively easy to avoid by timing it right. The seond factor is more complicated: I-24 through the Monteagle Pass between Nashville and Chattanooga. That stretch of road is often described as the most dangerous bit of interstate highway in the continental US. The grades through the pass are very long, very twisty, and very steep, and trucks often go out of control there if the road is wet. If you happen to be in front of one when it does, it could be deadly. Also, the pass is VERY prone to dense fog between about 10 pm and 8 am; don't try to cross it in the wee hours unless you feel comfortable with the possibility driving twisty mountain roads in near-zero visibility. That said, if you go through the Pass in the middle of a dry, sunny day you should be fine.
What I would say is that for a trip this week, study the weather along the route, and if snow or freezing rain is predicted in the mountains of Tennessee, hang a right at Nashville and make for Birmingham; the extra 90 minutes will be worth it to avoid those conditions. (Also, the Alabama route does bring you further South faster, which can be nice if you are running away from winter. Around half the trip is south of Montgomery, where it is usually much warmer than here. Last March I overnighted in Montgomery and was able to switch from a sweater, jeans and boots to shorts, sandals and a tee for the entire second half of the drive. The mental joy of that was like an extra day of vacation.)