> A* is not solving a different problem.

A* finds the shortest path from a node to a single other node. Dijkstra's finds the shortest paths from a node to all other nodes. If you use it as a search algorithm to find the shortest path to a single target, then yes, it's equivalent to A* with h(x)=0, but you're terminating Dijkstra's early (once your target is found) and not running the full algorithm.

A different problem in the sense that A* is useless (it degrades to dijkstra's) when there is no admissible heuristic. So I think it's reasonable to say that A* solves a different problem (namely, path finding when there is an admissible heuristic), since when there's no admissible heuristic it is identical to dijkstra's.