Separation of variables can be a powerful technique for solving many of the partial differential equations that arise in physics contexts. Upper-division physics students encounter this technique in multiple topical areas including electrostatics and quantum mechanics. To better understand the difficulties students encounter when utilizing the separation of variables technique, we examined students' responses to midterm exam questions and a standardized conceptual assessment and conducted think-aloud, problem-solving interviews. Our analysis was guided by an analytical framework that focuses on how students activate, construct, execute, and reflect on the separation of variables technique when solving physics problems. Here we focus on student difficulties with separation of variables as a technique to solve Laplace's equation in both Cartesian and spherical coordinates in the context of junior-level electrostatics. Challenges include recognizing when separation of variables is the appropriate tool, recalling or justifying the separated form of the potential and the need for the infinite sum, identifying implicit boundary conditions, and spontaneously reflecting on their solutions. Moreover, the type and frequency of errors was often different for separation of variables problems in Cartesian and spherical geometries. We also briefly discuss implication of these findings for instruction.