Q:

What's the difference between best-first search and A* search?

In my text book I noticed that both these algorithms work almost exactly the same, I am trying to understand what's the major difference between them.

The textbook traversed this example using A* the same way it did with best-first search.
Any help would be appreciated.

A:

Best-first search algorithm visits next state based on heuristics function f(n) = h with lowest heuristic value (often called greedy). It doesn't consider cost of the path to that particular state. All it cares about is that which next state from the current state has lowest heuristics.
A* search algorithm visits next state based on heristics f(n) = h + g where h component is same heuristics applied as in Best-first search but g component is path from the initial state to the particular state. Therefore it doesn't chooses next state only with lowest heuristics value but one that gives lowest value when considering it's heuristics and cost of getting to that state.

In your example above when you start from Arad you can go either
  straight to Sibiu (253km) or to the Zerind(374km) or Timisoara(329km).
  In this case both algorithms choose Sibiu as it has lower value f(n) =
  253.
Now you can expand to either state back to Arad(366km) or
  Oradea(380km) or Faragas(178km) or Rimnicu Vilcea(193km). For best
  first search Faragas will have lowest f(n) = 178 but A* will
  have Rimnicu Vilcea f(n) = 220 + 193 = 413 where 220 is cost of
  getting to Rimnicu from Arad (140+80) and 193 is from Rimnicu to
  Bucharest but for Faragas it will be more as f(n) = 239 + 178 = 417.

So now clearly you can see best-first is greedy algorithm because it would choose state with lower heuristics but higher overall cost as it doesn't consider cost of getting to that state from initial state

