Tracking Area Update procedure is used between MME and UE NAS. This way MME maintains the geographical proximity of UE in terms of tracking area(s). It helps MME to identify eNodeBs to trigger paging for an UE. There are two types of tracking area update procedures. One is known as "Periodic Tracking Area" and the other one is "Event based". Periodic tracking area update is triggered when UE moves to ECM-IDLE state and tracking area update timer T3412 is configured with some value. It means UE is connected to network but it is in IDLE mode.
In other case when UE is in ECM-IDLE state and it is not running periodic TAU timer, it will trigger TAU update if it enters into a tracking area which is not part of current TAI list.
Now coming to your question "TAU in connected mode". From my knowledge it is trigger during the handover only. When an UE moves from one eNodeB to another eNodeB as part of handover and the target eNodeB is part of different TA. UE triggers TAU procedure once the handover procedure gets completed.