2.5 or 3 way; why one over the other?

Answer is simple , if specification say the use woofer , midrange, and
tweeter , this is 3 way,  If specification say they use 3 drivers ,but
mention is 2 woofer with different cross point, or one woofer but
2 tweeters with different cross point , or regular tweeter and sypertweeter . this is 2.5 way. another configuration -woofer , widerange
 with no lowpass crossover , and sypertweeter is also considered 2.5 way,    Some speaker designers looking for something different
configuration ,but is better or not you have to decide using you best
tool is you ear.   But designers is not looking to make cheaper , just different. 

The answer requested is not really discernible from these varied answers. If different - how does ACTUALLY MANIFEST itself. They make different but why - THEY must have reason presumably, but joe public on this list does not know

The implementation is far more important than 2, 2.5, or 3. 

+1 @wcfeil.   

Well said and the bottom line truth of the matter.  

The electronic  and philosophy  is different