A* is not just that. The way it is said seems to imply that it is equal to Best First Search. Best First Search only relies on the heuristic function, so for the following graph, where A is the start node and h is the heuristic function:

+---A---+

|...........|

B h:1....E h:4

|

C h:2

|

D h:3

The traversal order according to Best First Search would be:

{A, B, C, D, E}

While A* will search by g(x) = h(x) plus d(x), the latter being the distance function from the start note to the given node. So, assuming the distance function to be equal to the depth of the node in the tree it would traverse it this way:

{A, B, C, E, D}

g(D) = h(D) + d(D) = 3 + 3 while g(E) = h(E) + d(E) = 4 + 1 = 5, so E goes first.