I am a little confused, the Wordpress 2.8 Widget API reffers to WordPress version 2.8 or is refferred to API version?
Should I use 2.8 Widget API for writing my WordPress 3.1.4 widget?
WordPress APIs usually refer to group of functions and concepts and are not versioned. Any versions refer to WP itself (or in some cases to bundled components, developed by third parties such as jQuery).
In WordPress version 2.8 creating widgets was refactored from older code (which you shouldn't care about) to newer class-based code (which you should use).
So it's not like it is exactly same as it was in 2.8 (likely it isn't) but approach is same since then.