As a initial phase of development we are need to perform API but why API automation is required after the product gone live. Is there any specific reason.
Because functional automation should be sufficient to track the functionality of the web site.
How does functional testing differ from API testing?
Functional testing focuses on validating software features and behavior (usually against specification documents generated by business and development teams). This black-box testing is often performed from the end-user perspective, and it can also include usability and UI testing in addition to feature validation.
API testing focuses on testing an application programming interface, which--at a high level--allows software systems to communicate. Instead of testing from an end-user point-of-view, the testing focuses on the mechanisms of the API. Generally, this involves making calls to the API and validating the response (e.g. returning the appropriate status code, correctly modifying a resource, etc).
Obviously, this is a superficial summary. But the takeaway is this: with functional testing, you're testing how the application behaves; with API testing, you're testing the internal mechanisms that drive application behavior.