That's a hard question because you really comparing apples and oranges.
The mini-max algorithm is useful in very specific situations. You need to have a well-defined state space (game board and pieces), well-defined state transitions (game rules), adversarial gameplay (players have opposing goals), low branching factor (few good moves in a given position) alternating turns, and a good static evaluation function (who has the most pieces). When these conditions exist, mini-max and its various heuristics (which includes alpha-beta pruning) is an efficient way to conduct a brute force search through the move tree.
A neural network is an artificial intelligence technique that in some ways mimics the human brain. The network learns by being exposed to a large body of input and being "rewarded" or "punished" if its response is right or wrong. It is a very general technique applicable to a wide variety of problems but usually does not give as good results as a special purpose solution.
The two are so different, comparing them is like asking what is a better food, grain or beef Wellington? Grain feeds the world, but you are unlikely to order it in a fancy restaurant.
To add to /u/kouhoutek's answer: they are very different tools that are implemented with very different structures - alpha-beta pruning uses a search tree, while a NN uses affine transformations composed with non-affine transformations --- its really a big composition of functions. These affine transformations are stored in multi-dimensional arrays and are often called (incorrectly) 'tensors'.
How this plays out in how they function: alpha-beta pruning itteratively eliminates less than optimal subtrees.
NN's often have massive numbers of parameters (100k's to millions), each of which gets "tuned" or adjusted itteratively.