Laser sources for future Gravitational Wave Detectors (GWDs) must meet demanding requirements including single-frequency output powers of above 700 W at 1064 nm, low noise and linear polarization with high beam quality and low higher-order mode content. Nd:YVO4 is an excellent material to the amplify the output of a low power seed, as 195W output power at 1064 nm with low noise and linear polarization have already been demonstrated in multi-stage Nd:YVO4 amplification. However, further power scaling was limited because of higher-order modes originating from aberrated thermal lensing. In this work, the aberrations of the thermal lens in Nd:YVO4 were analyzed in a single-stage amplifier configuration. The crystal was seeded and pumped at 1064 nm and 878.6 nm, respectively, while probing the thermal lens with a beam at 976 nm. The wavefront of this probe beam was analyzed with a Shack-Hartmann sensor. The amplifier was characterized up to 43W output power with 46\% extraction efficiency. We report a wavefront analysis with major contributions from defocus, astigmatism, and spherical aberration. The experimental results were complemented by an in-house developed numerical thermo-optical simulation model that, for the first time, included the major temperature-dependencies, i.e., of the emission cross-sections, thermal conductivity, thermal expansion, and heat capacity. We achieved excellent agreement of both output power and aberrations between simulations and experiment. Moreover, we introduced measures to compensate the aberrations in Nd:YVO4 leading the path to full compatibility with the demanded GWD requirements.