  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 {\it 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.
