As far as I know both T and F both mean triply degenerated, but F is used for vibrations and T is used in case of orbitals. So if T or F is used in case of vibrations their is no real difference, it is just a matter of habit.
The "g" comes from the German "gerade" and the "u" from the German "ungerade". It means that there is a center of inversion as symmetry element and g is symmetric to this center of inversion. Accordingly the rule of mutual exclusion holds - g modes can only be Raman-active and u modes only IR-active.