The coprecipitation method and the hydrothermal method use the same salts. Therefore, the two methods will have the same elements. It is impossible to talk about impurities of elements. The main synthesis route is also the same. By changing the pH, a mixture of hydroxides is obtained, which then form a complex spinel structure. In coprecipitation, only temperature and time are used to form the structure, while in hydrothermal synthesis, temperature and pressure are used. Pressure better forms the interpenetration of two oxides, the size and shape of the nanoparticles.
Besides the aforementioned issues, you should considered that for the solvothermal method you will need couple of more chemicals such as ethylene glycol and sodium acetate to add to the reaction medium!