Okay, I think I get it now -- though I still don't understand why "result34" in the OP should have come out your way rather than my way (given that the two input strings were completely identical).

Anyway, it seems like the algorithm you're after is better described as: align two strings s1,s2 such that a maximal final substring of s1 is an exact match to the initial substring of s2, and return the union of the two strings -- i.e. the common part preceded by the initial unmatched part (if any) of s1 and followed by the final unmatched part (if any) of s2.

Given that description, I think it could be simple: start with the first character of s2 aligned to the last character of s1, and shift the alignment one slot at a time toward the beginning of s1. At each iteration, if all overlapping characters match, store the resulting "union" string. At the point where the end of s1 is aligned with the end of s2, you're done -- the last union you stored was the best match possible.

(Or maybe I still really don't get it?)