We present the first comparison of observed stellar continuum spectra of high-redshift galaxies and mock galaxy spectra generated from hydrodynamical simulations. The mock spectra are produced from the IllustrisTNG TNG100 simulation combined with stellar population models and take into account dust attenuation and realistic observational effects (aperture effects and noise). We compare the simulated $D_n4000$ and EW(H$\delta$) of galaxies with $10.5 \leq \log(M_\ast/M_\odot) \leq 11.5$ at $0.6 \leq z \leq 1.0$ to the observed distributions from the LEGA-C survey. TNG100 globally reproduces the observed distributions of spectral indices, implying that the age distribution of galaxies in TNG100 is generally realistic. Yet there are small but significant differences. For old galaxies, TNG100 shows small $D_n4000$ when compared to LEGA-C, while LEGA-C galaxies have larger EW(H$\delta$) at fixed $D_n4000$. There are several possible explanations: 1) LEGA-C galaxies have overall older ages combined with small contributions (a few percent in mass) from younger ($