Inductive Logic Progamming (ILP) is a class of machine learning algorithms that learn logic programs from examples and background knowledge (where background knowledge is a logic theory, i.e. another logic program, as ar the examples).
SAT is the boolean satisfiability problem, the problem of finding variable assignments to the variables of a boolean formula that make the formula true.
So the two are not similar and one is not an instance of the other.
The δILP paper describes a differentiable ILP system that learns logic programs with a differentiable logic representation.
Finally, the paper above (SATNet) describes a differentiable satisfiabiilty solver that can be incorporated in a neural network architecture to enable the nerual net to solve satisfiability problems and perform reasoning.
I meant was ASP is a higher level 'interface' to SAT, sorry for the confusion. Work done at Imperial by Law and Evans in ILP compiles down to SAT solving via ASP. IDK if δILP goes to SAT.