To answer your specific question: absolutely none.
The Millenium run is a "dark matter-only" simulation. In this sort of simulation gas physics is taken to play a negligible role. All the gas (and stars, indeed all "baryonic matter" as it's called in the jargon) is removed and replaced with additional dark matter. The extra dark matter is added just to keep the density of the Universe the same as it would have been with the gas included. On some scales, the behavior of the dark matter is not very sensitive to the processes of "baryon physics", so this process ends up giving a simulation output that is a decent approximation to the large-scale structure of the Universe. It's also much cheaper in terms of computing time to run a dark matter-only simulation than one that includes gas physics.
Smoothed particle hydrodynamics is, as the name implies, a technique to solve the equations of hydrodynamics, so it is only used in simulations that model gas (stars are usually treated as a collisionless fluid, or if resolution is sufficient, as individual point masses, and so are treated separately). In SPH, the gas is discretized as a collection of Lagrangian "particles". Each particle has a position and velocity, and carries fluid properties (e.g. temperature, pressure, etc.) as well. To obtain the properties of the fluid at an arbitrary point in space, the appropriate property is calculated as a kernel-smoothed average over nearby particles. Typically the nearest particles carry the most weight, and particles beyond a certain distance don't contribute at all. These "smoothed" values are the values used in solving the equations of hydrodynamics (Navier-Stokes equations, or some simplification thereof), which yields the forces acting on each particle and the energy to be exchanged between particles, allowing integration of the system.
Recent cosmological simulations have begun to include hydrodynamics. The two currently most relevant are EAGLE, which uses SPH (modified P-Gadget-3 code), and Illustris, which uses an entirely different technique called a moving mesh (AREPO code) to solve the hydrodynamics equations.