You might also ask if there's any particular reason why we would use a neural net. If we're to train a neural net to play chess, we need to be able to:
1. Feed it positions as input vectors (easy enough),
2. Decide on an output format. Perhaps a distribution over possible moves (but then, how to represent that such that the meaning of a specific output cell doesn't change drastically based on the board state? Or perhaps instead, we let the resulting board state after a candidate move be the input, and let the output be a score that represents the desirability of that state. That'll require exponentially more forward/backprop passes, though.
3. Provide it with an error signal to whatever output vector it produces. This is the really tricky bit, since we don't know whether a given move will result in victory until the very end.
Do we play the game to the very end, storing decisions as we go, and then at the end, replay each input, feeding it an error signal if we lost? This will give the same error to the good moves as to the ones that actually lost the game. With enough games, this will work, since the good moves will get positive feedback a bit more often than negative, and vice versa for the bad ones. But it'll take a lot of games. More than a human is going to be willing to play. We can have different networks learn by playing against each other, but not on 1996 hardware.
Do we instead provide a score based on another heuristic of the board state? In that case, why not just use minimax? It's provably optimal for a given heuristic up to however many moves deep we look, and it doesn't need training.
Add to this the fact that if we don't choose a good representation at each of these steps, there's a good chance that the network will only learn the positions it's specifically been trained on, rather than generalizing to unseen states, which is the main reason for using a neural network in the first place.
It's certainly possible to use neural nets to learn chess (DeepMind's approach can be found here, for instance), but they're not a natural fit to the problem by any means. Minimax, by contrast, fits the problem very well, which is why it was one of the techniques used by Deep Blue. Neural nets are an amazing tool, but they're not always the right tool for the job.
I didn't stress this point much, since K.C. already brought it up, but training large neural nets require us to perform a huge number of matrix vector multiplications, and this wasn't especially practical before GPUs got powerful and cheap.