tbupdater and Core updater are two different modules.
tbupdater updated the modules up to thirtybees version 1.4, now this functionality is included in the core.
Core updater updates everything else and it should continue to be used. This module is compatible with all thirtybees versions since its introduction back in 1.0.8 or something like that...
If you updated to thirtybees 1.5 you should check the forum thread, there are couple of other such minor changes that will take you 5 minutes - minification module and mail transport module and configuration.