We can use Venn diagram to solve such questions.
$S_1$ and $S_2$ are two disjoint regions in the Venn diagram which can't be compared.
We can also solve them as
$S_1 = ((A \cap B) - (B \cap C)) - C$
$\qquad = ((A \cap B) \cap \overline {(B \cap C)}) - C$
$\qquad = ((A \cap B) \cap (\overline {B} \cup \overline{C})) - C$
$\qquad = (A \cap B \cap \overline{C}) - C$
$\qquad = (A \cap B \cap \overline{C}) \cap \overline{C}$
$\qquad = A \cap B \cap \overline{C}\qquad \to (1)$
$S_2 = (A - (A - C)) - (A -B)$
$\qquad = (A - (A \cap \overline{C}) - (A - B)$
$\qquad = (A \cap \overline{(A \cap \overline{C})} - (A \cap \overline{B})$
$\qquad = (A \cap (\overline{A} \cup {C}) - (A \cap \overline{B})$
$\qquad = (A \cap {C}) - (A \cap \overline{B})$
$\qquad = (A \cap {C}) \cap \overline{(A \cap \overline{B})}$
$\qquad = (A \cap {C}) \cap (\overline{A} \cup {B})$
$\qquad = (A \cap {C} \cap {B})\qquad \to (2)$
$(1)$ and $(2) \implies S_1$ and $S_2$ have no subset relationship.
So, D is the answer.