First off, the more uncommon one. To pore (over) something is to be absorbed in the reading of it. “She was so busy poring over the article on semiotics that she missed her stop on the Tube”. A pore is also a minute opening in the skin (etc.) through which liquids and gases can pass. But I don’t think that’s the usage that gets mixed up with this pair.
To pour (over, out, etc.) is to flow or cause to flow in a steady stream. “Water poured down the wall as the shower leaked from above”; “she poured herself a cup of tea”.
Short but sweet. The same sound, but different spellings, meanings and, well, words.
“As she pored over the ancient manuscripts, it began to pour with rain”.