What are the differences between greedy best-first and the A* search algorithms? How is A* better than the greedy best-first search algorithm?
According to the book Artificial Intelligence: A Modern Approach (3rd edition), by Stuart Russel and Peter Norvig, specifically, section 3.5.1 Greedy best-first search (p. 92)
Greedy best-first search tries to expand the node that is closest to the goal, on the grounds that this is likely to lead to a solution quickly. Thus, it evaluates nodes by using just the heuristic function; that is, $f(n) = h(n)$.
In this same section, the authors give an example that shows that greedy best-first search is neither optimal nor complete.
In section 3.5.2 A* search: Minimizing the total estimated solution cost of the same book (p. 93), it states
A* search evaluates nodes by combining $g(n)$, the cost to reach the node, and $h(n)$, the cost to get from the node to the goal $$f(n) = g(n) + h(n).$$
Since $g(n)$ gives the path cost from the start node to node $n$, and $h(n)$ is the estimated cost of the cheapest path from $n$ to the goal, we have $f(n)$ = estimated cost of the cheapest solution through $n$.
Thus, if we are trying to find the cheapest solution, a reasonable thing to try first is the node with the lowest value of $g(n) + h(n)$. It turns out that this strategy is more than just reasonable: provided that the heuristic function $h(n)$ satisfies certain conditions, A* search is both complete and optimal. The algorithm is identical to uniform-cost search except that A* uses $g + h$ instead of $g$
What you said isn't totally wrong, but the A* algorithm becomes optimal and complete if the heuristic function h is admissible, which means that this function never overestimates the cost of reaching the goal. In that case, the A* algorithm is way better than the greedy search algorithm.