Some frames of reference are very close to being inertial frames, and others aren't. There's a big difference between inertial and non-inertial frames. In non-inertial frames, you have to introduce fictitious forces in order for the physics to work out.
In short, the universe cares about acceleration, but not constant motion. Accelerating frames are different from non-accelerating frames.
There is no center of the Universe. If the Universe is flat or negatively curved, then it is infinitely large, and if it is positively curved, then the volume is finite, but space is periodic.
Space is intrinsically expanding, and that expansion is accelerating. That's quite different from acceleration in Newtonian physics. The statements I made about acceleration are in the context of Newtonian, pre-Relativistic physics, and are only approximate in a Relativistic framework.
In short, the universe cares about acceleration, but not constant motion. Accelerating frames are different from non-accelerating frames.