Why is CuSO4 used as electrolyte while purifying copper?
During copper electrorefining, impure $\ce{Cu}$ from the anode is oxidized and dissolved, while $\ce{Cu^2+}$ is reduced at the cathode, forming the deposit of refined copper. Because anodic impurities which are less noble than copper – most notably $\ce{Ni}$, $\ce{As}$ and $\ce{Sb}$ – also dissolve in the electrolyte, the cell voltage is kept in the range of 0.2-0.4 V to avoid co-deposition of these metals at the cathode. 

Copper dissolution is the main oxidation half-reaction at the anode, and the oxidation of anions from the electrolyte needs to be minimized, since it is an unwanted side reaction which reduces current efficiency of the process. In a $\ce{CuSO4}$ electrolyte, this would be oxidation of $\ce{OH-}$ (from water dissociation, so concentration will be low) to $\ce{O2}$, while in a $\ce{CuCl2}$ solution, $\ce{Cl-}$ will be oxidized to chlorine gas, which would present a greater hazard because of its toxicity.

The electric conductivity of the electrolyte is another important factor, which varies with different copper salts and their concentrations. $\ce{CuCl2}$ and $\ce{CuSO4}$ are both readily soluble in water; however, the relative molar conductivities of these solutions will be different. In this study, a copper chloride electrolyte proved to be inferior to copper sulfate and copper nitrate in terms of deposit quality and stability of cell potential.