Numerical and analytical studies of decaying, two-dimensional (2D) Navier-Stokes (NS) turbulence at high Reynolds numbers are reported. The effort is to determine computable distinctions between two different formulations of maximum entropy predictions for the decayed, late-time state. Though these predictions may be thought to apply only to the ideal Euler equations, there have been surprising and imperfectly-understood correspondences between the long-time computations of decaying states of NS flows and the results of the maximum-entropy analyses. Both formulations define an entropy through a somewhat ad hoc discretization of vorticity to the "particles" of which statistical mechanical methods are employed to define an entropy, before passing to a mean-field limit. In one case, the particles are delta-function parallel "line" vortices ("points" in two dimensions), and in the other, they are finite-area, mutually-exclusive convected "patches" of vorticity which in the limit of zero area become "points." The former are taken to obey Boltzmann statistics and the latter, Lynden-Bell statistics. Clearly, there is no unique way to reach a continuous differentiable vorticity distribution as a mean-field limit by either method. The simplest method of taking equal-strength points and equal-strength, equal-area patches is chosen here, no reason being apparent for attempting anything more complicated. In both cases, a non-linear partial differential equation results for the stream function of the "most probable" or maximum-entropy state, compatible with conserved total energy and positive and negative vorticity fluxes. These amount to generalizations of the "sinh-Poisson" equation which has become familiar from the "point" formulation. They have many solutions, which must be obtained numerically, and only one of which maximizes the entropy on the basis of which it was derived. These predictions can differ for the point and patch discretizations. The intent here is to use time-dependent, spectral-method direct numerical simulation of the Navier-Stokes equations to see if initial conditions which should relax to different late-time states under the two formulations actually do so.