We present measurements of the E-mode polarization angular auto-power spectrum (EE) and temperature-E-mode cross-power spectrum (TE) of the cosmic microwave background (CMB) using 150 GHz data from three seasons of SPTpol observations. We report the power spectra over the spherical harmonic multipole range and detect nine acoustic peaks in the EE spectrum with high signal-to-noise ratio. These measurements are the most sensitive to date of the EE and TE power spectra at and , respectively. The observations cover 500 , a fivefold increase in area compared to previous SPTpol analyses, which increases our sensitivity to the photon diffusion damping tail of the CMB power spectra enabling tighter constraints on ΛCDM model extensions. After masking all sources with unpolarized flux mJy, we place a 95% confidence upper limit on residual polarized point-source power of at , suggesting that the EE damping tail dominates foregrounds to at least with modest source masking. We find that the SPTpol data set is in mild tension with the ΛCDM model ( ), and different data splits prefer parameter values that differ at the level. When fitting SPTpol data at , we find cosmological parameter constraints consistent with those for Planck temperature. Including SPTpol data at results in a preference for a higher value of the expansion rate ( ) and a lower value for present-day density fluctuations ( ).