You can also find infinitely many smooth compactifications of $S^2 \times S^2 \setminus \Delta$ which are pairwise homeomorphic but not diffeomorphic.

In fact, the complex quadric $\mathbb{CP}^1 \times \mathbb{CP}^1$ blown-up in one point is isomorphic to $\mathbb{CP}^2$ blown-up in two points; therefore $\mathbb{CP}^1 \times \mathbb{CP}^1$ blown-up in four points is isomorphic to $\mathbb{CP^2}$ blown-up in five points.

But the topological 4-manifold $\mathbb{CP}^2 \sharp 5 \overline{\mathbb{CP}^2}$ supports infinitely many different smooth structures, see [Park-Stipsics-Szabo, Exotic smooth structures on $\mathbb{CP}^2 \sharp 5 \overline{\mathbb{CP}^2}$, Math. Research Letters 12 (2005)], and this proves the assertion.

Using the nice answer of Scott Carnahan, you can also prove that there are find infinitely many smooth compactifications of $S^2 \times S^2 \setminus \Delta$ which are pairwise homeomorphic but not diffeomorphic.

In fact, the complex quadric $\mathbb{CP}^1 \times \mathbb{CP}^1$ blown-up in one point is isomorphic to $\mathbb{CP}^2$ blown-up in two points; therefore $\mathbb{CP}^1 \times \mathbb{CP}^1$ blown-up in four points is isomorphic to $\mathbb{CP^2}$ blown-up in five points.

But the topological 4-manifold $\mathbb{CP}^2 \sharp 5 \overline{\mathbb{CP}^2}$ supports infinitely many different smooth structures, see [Park-Stipsics-Szabo, Exotic smooth structures on $\mathbb{CP}^2 \sharp 5 \overline{\mathbb{CP}^2}$, Math. Research Letters 12 (2005)], and this proves the assertion.