A CNC lathe would typically have a simple manual toolpost like an Aloris, whereas a turning center would typically have an automatic programmable turret that would hold 8 to 12..or more....tools. There is a huge difference in productive capability to have that programmable turret, hence the "turning center" moniker.
For even higher production, there could be a second turret, a sub spindle (you can turn both ends of the part without turning it around), and live turret stations (for milling groves, flats, hex shapes, or whatever, in the turned parts)..still a "turning center", but sometimes a "machining cell" or "high productive cell" when it can "do it all"
Other differnces are that a CNC lathe is typically set up similar to an engine lathe as far as basic configuration of ways, whereas a turning center would typically have a slanted bed.