All three of the trap blocks (Kuri-Tsuke, Kuri-Hanashi & Tai-Atari) all begin the same way: an attempt to execute Kuri-Tsuke.
It is because of uke’s response that the second & third trap blocks happen. Tori’s actions always begin the same way, then adjust to uke’s response.
In Kuri-Tsuke, the initial trap action is successful. Uke initiates movement with this right foot in order to escape the trap.
In Kuri-Hanashi, the initial trap action is unsuccessful. Uke has seen this trap before, and steps out with his right foot at the peak of tori’s upward action to thwart the trap action. Or perhaps uke reacts earlier than usual, or maybe even tori reacts later than usual.
In Tai-Atari, the initial trap action is unsuccessful. Uke has seen the initial trap before, and steps with his left foot in an effort to attain a position to cut tori.
Each of these response actions from uke has a a very different feel. It is important that tori sincerely attempt to execute Kuri-Tsuke, and only because of uke’s actions do the other two ever materialize.
In addition, tai-atari presents a concept in kata the other two don’t. When we see tai-atari occur in kata it is used to turn uke 180 degrees. Imagine starting with the sun in your eyes (as we see in katas #7, #9 and #11) and being able to turn the tables so that uke has the sun in his eyes.