Determining the existence of a forced win vs determining the best outcome
They are equivalent under Turing reductions, assuming that the game has finite/polynomial-size branching factor (i.e., only that many moves are possible from each position).  I don't know if they are equivalent under Karp reductions.

Consider a position $P$.  Assume there are $k$ moves that are legal there.  Let $P_1,\dots,P_k$ denote the possible positions after making a single move.

Check whether $P$ is a forced win for player #1.  If yes, you are done.

Check whether any of $P_1,\dots,P_k$ is a forced win for player #2.  If they are all a forced win for player #2, then $P$ is a forced lose for player #1.

Otherwise, $P$ is a forced draw for player #1.