We compare the properties of ordinary strong magnetohydrodynamic (MHD) turbulence in a strongly magnetized medium with the recently discovered viscosity-damped regime. We focus on energy spectra, anisotropy, and intermittency. Our most surprising conclusion is that in ordinary strong MHD turbulence, the velocity and magnetic fields show different high-order structure function scalings. Moreover, this scaling depends on whether the intermittency is viewed in a global or local system of reference. This reconciles seemingly contradictory earlier results. On the other hand, the intermittency scaling for viscosity-damped turbulence is very different and difficult to understand in terms of the usual phenomenological models for intermittency in turbulence. Our remaining results are in reasonable agreement with expectations. First, we find that our high-resolution simulations for ordinary MHD turbulence show that the energy spectra are compatible with a Kolmogorov spectrum, while viscosity-damped turbulence shows a shallow k^-1 spectrum for the magnetic fluctuations. Second, a new numerical technique confirms that ordinary MHD turbulence exhibits Goldreich-Sridhar type anisotropy, while viscosity-damped MHD turbulence shows extremely anisotropic eddy structures. Finally, we show that many properties of incompressible turbulence for both the ordinary and viscosity-damped regimes carry over to the case of compressible turbulence.