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A* expands on a node only if it seems promising. It's only focus is to reach the goal node as quickly as possible from the current node, not to try and reach every other node.
A* is complete, which means that it will always find a solution if it exists.
Pro Can be morphed into other algorithms
A* can be morphed into another path-finding algorithm by simply playing with the heuristics it uses and how it evaluates each node. This can be done to simulate Dijkstra, Best First Search, Breadth First Search and Depth First Search.
Con Not useful if you have many target nodes
If you have many target nodes and you don't know which one is closest to the main one, A* is not very optimal. This is because it needs to be run several times (once per target node) in order to get to all of them.