Fitting whole spectra at intermediate spectral resolution (R = 1000 - 3000), to derive physical properties of stellar populations, appears as an optimized alternative to methods based on spectrophotometric indices: it uses all the redundant information contained in the signal. This paper addresses the validation of the method and it investigates the quality of the population models together with the reliability of the fitting procedures. Our method compares observed optical spectra with models to derive the age, metallicity, and line broadening due to the internal kinematics. It is insensitive to the shape of the continuum and the results are consistent with Lick indices but three times more precise. We are using two algorithms: STECKMAP, a non-parametric regularized program and NBURSTS a parametric non-linear minimization. We compare three spectral synthesis models for single stellar populations: Pegase-HR, Galaxev (BC03) and Vazdekis/Miles, and we analyse spectra of Galactic clusters whose populations are known from studies of color-magnitude diagrams (CMD) and spectroscopy of individual stars. We find that: (1) The quality of the models critically depends on the stellar library they use, and in particular on its coverage in age, metallicity and surface gravity of the stars. Pegase-HR and Vazdekis/Miles are consistent, while the comparison between Pegase-HR and BC03 shows some systematics reflecting the limitations of the stellar library (STELIB) used to generate the latter models; (2) The two fitting programs are consistent; (3) For globular clusters and M67 spectra, the method restitutes metallicities in agreement with spectroscopy of stars with a precision of 0.14 dex; (4) The spectroscopic ages are very sensitive to the presence of a blue horizontal branch (BHB) or of blue stragglers. A BHB morphology results in a young SSP-equivalent age. Fitting a free amount of blue stars in addition to the SSP model to mimic the BHB improves and stabilizes the fit and restores ages in agreement with CMDs studies. This method is potentially able to disentangle age or BHB effects in extragalactic clusters. Full spectrum fitting is a reliable method to derive the parameters of a stellar population.