WHY NOT USE A LOOK-UP TABLE INSTEAD OF A NETWORK?

Because a look-up table would be extremely inefficient. A look-up table requires $k^{n+1}$ entries for all the conditional probabilities corresponding to all possible combinations of $n$ previous characters and possible next characters. In addition, a special procedure is required for dealing with previously unseen combinations of input characters. In contrast, the size of a neural net typically grows in proportion to $n^2$ (assuming the number of hidden units grows in proportion to the number of input units), and its inherent ``generalization capability'' is going to take care of previously unseen combinations of input characters (hopefully by coming up with good predicted probabilities).