What the difference in configs ?
Project with Branches Tab:
Project without Branches Tab:
If you only have one branch specified in the VCS config, you won't see a Branches tab.
TeamCity doesn't show branches in a project in case it knows nothing about them, i.e. there are no builds/changes in the project since project have been created. So if there is an old project with builds, newly created one has no knowledge regarding those builds. Try running a build in a branch, it should help.
Try to add branch specification *
in VCS Root settings and test connection to force Team City to fetch information about branches from git repository.