Folks, I'm new to this field and trying to better understand, when does it make sense to go with the Teradata's solution and when does it make sense to go with the Exadata's solution. Any opinions will be greatly appreciated.

The difference depends on the nature of the problem you are facing and the tools that you have available to solve it. Without knowing more about the specifics of your situation, there isn't a good way to make a useful recommendation.

-PatP