Why we need to print and proof (or why DO we?)

Almost all of the average person's writing is done digitally nowadays. Yet our brains are still to become wired with the ability to pick up errors when reading on the screen. I don't mean just spelling or grammatical errors, what else would the computer checks be for? (and I am very thankful my email is set to auto check - best thing I ever discovered).
With anything important, or anything that will be read by others with discerning eyes, I always default to printing and reading the hard copy. And I always find errors that I do not pick up on the computer. Like the glaring one here on the cover sheet of my first assignment, due tomorrow,  Monday August 8. Not to be confused with Monday August 7 - duh.
Anyway, wish me luck, I have hopefully saved a correctly corrected copy, and am about to wave good-bye to it, and 'post' it to my lecturer. I will not read it again, as I am sure I will pick up other nuances that I will want to alter, But hopefully I have spotted all the mistakes!