Exactly what the question says;
If all the protons and electrons in every single atom in the universe were swapped for their anti-particles, what would essentially change?
Changing all particles into antiparticles and vice versa is known as a charge symmetry operation (C) and for a long time it was believed this would leave everything totally unchanged. Similarly mirroring the universe (a parity symmetry operation, P) would seem to leave everything unchanged. However, in 1956 it was found that there are weak interactions that do not obey parity symmetry - the left and right handed forms have different probability. It was believed that the combined CP symmetry was still true, but in 1964 that was also found to fail: CP violation.
So the antimatter universe would be very slightly different from ours. The effect is small, and only occurs in weak interactions involving nonzero strangeness number. That means that the effects in everyday physics will be very small: while normal protons and neutrons have a pinch of strange quark presence it is tiny. It would be felt most strongly for weak interactions - that are also just a small part of what is going on. Presumably it would change energy levels in nuclear transitions to a tiny degree, which may change a few fusion pathways in stars.
So there would be a difference, but it would likely be almost imperceptible, in the form of different elemental abundances of heavier elements.