Thinking in computer terms: "reading" is trying out variations. Because of limited time you cannot try all 361 points on the board for every move, not even all points in a corner. So one relies on learned patterns, intuition, heuristics etc. to know which potential moves should be investigated further (i.e. read out).

I have read somewhere that the difference between pros and amateurs is that pros spend most of their time evaluating positions after a sequence, whereas amateurs spend most of their time reading. Furthermore, on a typical move a professional only considers 3-4 potential moves.

I guess this is what is referred to "seeing": choosing promising moves among the potential candidates which to read out. This is where the phrase "I did not see that move" comes from: even if you can read 20 moves deep, you prune at (almost) every step and can miss a surprising unusual move at step 2. And sometimes you prune the wrong ones. So I see a difference between "reading" and "seeing".