We present a study of the X-ray properties of the nearby face-on Scd spiral galaxy M101 based on recent XMM-Newton observations. In this third and final paper in the present series, we focus on the spatial and spectral properties of the residual emission, after excluding bright X-ray sources with L(X) > 10^37 erg/s. Within a central region of radius 10 arcmins (21 kpc), the X-ray emission broadly traces the pattern of the spiral arms, establishing a strong link with recent star formation, but it also exhibits a radial scale length of \~2.6 arcmins (5.4 kpc) consistent with optical data. We estimate the soft X-ray luminosity within the central 5 arcmins (10.5 kpc) region to be L(X) ~2.1 x 10^39 erg/s (0.5-2 keV), the bulk of which appears to originate as diffuse emission. We find a two-temperature thermal model best fits the spectral data with derived temperatures of 0.20 +/- 0.01 keV and 0.68^(+0.06)_(-0.04) keV which are very typical of the diffuse components seen in other normal and starburst galaxies. More detailed investigation of the X-ray morphology reveals a strong correlation with images recorded in the far-UV through to V band, with the best match being with the U band. We interpret these results in terms of a clumpy thin-disk component which traces the spiral arms of M101 plus an extended lower-halo component with large filling factor.