Robust control is used for when you know that parameters are going to fall within particular bounds. For example, for a mass-produced robot arm, the arm masses fall between X and Y, the moments of inertia between I and J, etc. You don't know exactly what they are, but you can analyze the boundary conditions (i.e., root locus analysis) and ensure stability and performance parameters are met at all times.
Adaptive control is used for when you don't know that the parameters are going to fall within particular bounds. Maybe you've got a kite controller, but the wind can be anything between zero and a hurricane, or an autopilot system in an aircraft where the total passenger and equipment weight isn't known.
A robust controller is developed and analyzed once, and then the gains are fixed. An adaptive controller must constantly assess the system performance and attempt to adjust control gains on-the-fly. The adaptive controller must be a more computationally complex controller (for comparable systems) because it cannot leverage a priori knowledge like the robust controller.
Ultimately, the best controller is the one that performs the best for your system. I would argue that robust control would probably best be applied to robots working in controlled conditions, or fabricated of mass produced parts, and adaptive control then applied to robots working in uncertain environments.