T Tau stars display different X-ray properties depending on whether they are accreting (classical T Tau stars: CTTS) or not (weak-line T Tau stars: WTTS). We use data from the XMM-Newton Extended Survey of the Taurus Molecular Cloud (XEST) to study differences in X-ray properties between CTTS and WTTS. We perform correlation and regression analysis between X-ray parameters and stellar properties. We confirm the existence of a X-ray luminosity (Lx) vs. mass (M) relation, Lx ~ M^(1.69 +/- 0.11), but this relation is a consequence of X-ray saturation and a mass vs. bolometric luminosity (L*) relation for the TTS with an average age of 2.4 Myr. X-ray saturation indicates Lx = const L*, although the constant is different for the two subsamples: const = 10^(-3.73 +/- 0.05) for CTTS and const = 10^(-3.39 +/- 0.06) for WTTS. Given a similar L* distribution of both samples, the X-ray luminosity function also reflects a real X-ray deficiency in CTTS, by a factor of ~ 2 compared to WTTS. The average electron temperatures Tav are correlated with Lx in WTTS but not in CTTS: CTTS sources are on average hotter than WTTS sources. The most fundamental properties are the two saturation laws, indicating suppressed Lx for CTTS. We speculate that some of the accreting material in CTTS is cooling active regions to temperatures that may not significantly emit in the X-ray band, and if they do, high-resolution spectroscopy may be required to identify lines formed in such plasma, while CCD cameras do not detect these components. The similarity of the Lx vs. Tav dependencies in WTTS and main-sequence stars as well as their similar X-ray saturation laws suggests similar physical processes for the hot plasma, i.e., heating and radiation of a magnetic corona.