system
Given the following post, respond as if you're commenting on Reddit:
human
Would discovering that the universe was actually infinite or finite change anything regarding our understanding of physics?
gpt
If a particle is confined to a finite volume its allowed energies become discrete and quantized.  However, if the size of that volume increases to infinity the gap between different allowed energies approaches zero and you get a continuum.  Thus, one can say that the physics of a confined particle and a free particle are very different. 

However, this is only true if one is considering physics whose energy scale is much less than the energy gap associated with this confinement. For example, a system of particles at room temperature on Earth (~300 K) has ~0.025 eV (electronvolts, a unit of energy) of thermal kinetic energy. Thus, such a particle, like say an electron in a solid, will only behave "differently" than a free particle if you confine it so much that the discrete energy gaps become larger than 0.025 eV (which occurs for electrons in silicon if you confine it to a volume of less than ~10 nm, something that is happening in the trillions of transistors driving the computer you're using right now - electrons in MOSFET transistors exist in an effective 2D world because of this quantum confinement).  If you confine it to a larger box than ~10 nm, say so that energy gap is ~0.0001 eV (which is much smaller than the 0.025 eV of thermal energy) then it will behave like a free particle because the "fine detail" of its structure of allowed energy is hidden by the much larger fluctuations in energy brought about by the thermal motion; kind of like a microscope that isn't zoomed in enough so that a material we know is made of a complex structure of animal cells, just looks like skin.

So, coming back to your original question,  in an infinite universe "free" particles have continuous energy spectra, where in a finite universe (whether it be finite in the sense that it ends, or if it's periodic, where you loop back to where you start) energy spectra are discrete.  HOWEVER, this is only discernable if we consider phenomena whose energy scale is less than the discreteness. And the larger a finite universe is, the tinier this discretization detail is.

We have never observed "discreteness" of free particles, so either the universe is infinite, or it's at least big enough such that the discreteness is tinier than our ability to detect.  Which brings us to the pragmatic answer that if the discreteness ends up being tinier than an energy scale we could EVER, even in principle, be able to discern, then it pragmatically makes no difference.  Everything will effectively behave as if it were a continuum anyway. 

Realistically, the universe that is currently observable is already far, far, far, far, far too big, such that even if there was magically an "end" just beyond it (or a periodic looping), the energy discretization would still be so outrageously tiny we'd never tell the difference.  So realistically: no, it won't matter to Earth physics.

However, it would tell us something crucial about its curvature and the type of cosmology that describes our universe.  Thus, it would tell us something about the fate of the universe.