Pretty sure Memphis has the warmer winters. Both the USDA Hardiness Zone Map and the Arborday Hardiness Zone Map show Memphis as having warmer winter temps.

BUT...be aware that local micro and mesoclimates can make a big difference, and probably more difference than the average difference between the two cities. In other words, somewhere in Chattanooga is almost certainly warmer than the average temps in Memphis, and vice versa. Low lying areas (cold pockets), for example, tend to be colder than surrounding higher areas; areas with a lot of rock may stay warmer than an area without rock; shady areas will be colder than sunny areas; etc.