Hi Gayathri:
As a starting point to understand the concepts of panchangam, start with this link -
http://en.wikipedia.org/wiki/Ayanamsa.
If you understand the concept of "Ayanamsa", you will then be able to take the next step in understanding the differences between different panchangams. Start with the link above and use google.
I din't know anything about Vakiya and Thirukanitha panchangams but a search on the web landed at the page below.
http://in.answers.yahoo.com/question/index?qid=20080428100441AAcXLV7.
You will be able to make more sense out of it slowly as you grasp the concept of "Ayanamsa".
It might be overwhelming but take one step at a time and be patient. I had the luxury of my father explaining me the basics of Panchangam. Strong roots in the science of Panchangam will take you a long way in learning Astrology.
Good luck! And keep asking questions.
-Sagar