The interaction between two spherical particles coated by end-grafted polymers immersed in a good solvent is examined using the strong-stretching theory of Milner, Witten, and Cates. Our calculation allows the polymer chains to tilt outward from the axis of closest approach, where the separation between the substrates is a minimum. This transfers the stress toward the extremity of the contact region, but to such a degree that the central chains can experience a reduction in free energy relative to the unperturbed brush. Nevertheless, the chain tilt has a small effect on the overall free energy, and thus the traditional Derjaguin approximation predicts a reasonably accurate interaction potential, provided the particles are large relative to the height of their brushes. Better yet, we propose an improved version that performs well for particles not much larger than the brushes.