It looks like this is a more comprehensive version of FlashFill (can do more tasks), and it is based on deep learning instead of previous rule-based techniques in FlashFill.
Grammatical inference learns a grammar from a set of examples, where here it seems the paper is learning a program (a derivation in the grammar) from examples.
Which Dawn Song paper are you talking about here? I think among all the recent approaches proposed recently on neural program induction, this is the first one that is end-to-end trained and learns only from input-output examples without any hacks!