Astrometric missions like Gaia provide exceptionally precise measurements of stellar positions, parallaxes, and proper motions. Gravitational waves traveling between the observer and distant stars can induce small, correlated shifts in their apparent positions, a phenomenon known as astrometric deflection. The precision and scale of astrometric datasets make them well suited for searching for a stochastic gravitational-wave background, whose signature appears in the two-point correlation function of the deflection field across the sky. In space-based astrometry, the ultimate sensitivity of such measurements is reduced by systematic uncertainties in the satellite's absolute attitude reconstruction, which constrain the accuracy of absolute astrometry. These orientation errors can be mitigated by focusing on relative angular separation between pairs of stars, which effectively cancel out common-mode orientation noise. In this work, we compute the astrometric response and the overlap reduction functions for this differential approach, correcting previous expressions presented in the literature. We use a Fisher matrix analysis to compare the sensitivity of differential astrometry to that of conventional absolute astrometry. Our analysis shows that while the differential method is theoretically sound, its sensitivity is limited for closely spaced star pairs. Pairs with large angular separations provide competitive sensitivity, but, when considered in connection with Gaia, it is unclear whether the differential strategy would be effective, since instrumental systematics are not expected to remain correlated on such scales. Finally, we demonstrate that combining astrometric data with observations from pulsar-timing arrays leads to slight improvements in sensitivity at frequencies greater than or similar to 10-7Hz.
