Why is sugar water sticky, but salt water is not?

Not entirely sure, but I reckon it might have something to do with sugar being a carbohydrate, like starches (which are, of course, sticky/gummy).

The fact that both are crystalline solids belies deeper structural differences.

My chemistry classes are quite in the past, but I think it is because when salt dissolves, its crystal is being broken apart into chlorine and natrium ions, while sugar retains its long chained molecules, which are likely to interlock and thus result in stickiness.

That’s right! And I can show you soap in crystalized form, which never will stick, and I can show you a very strong glue in crystalized form wich is sticky like hell. So it’s not about what form the material has, it’s about what is that you want to crystallize!

Salt water can feel sticky.

@ragingloli is correct, although most sugars are actually not long-chained molecules (unlike starch which is a carbohydrate consisting of a large number of glucose units joined together by glycosidic bonds).