I love this story about the difference between memorizing things "by rote" versus "by heart". Although I don't see much reference to Shakespeare in the text, it's not hard to extrapolate. How often do we use the tired old example of your high school English teacher who forced you to memorize, just for the sake of memorization, the balcony scene? And the generation of students that can recite it but still hate it, or worse, hate it all the more because of that?
So there's a thought that if you learn by heart it means you take the poem right into yourself, it becomes part of you. And it remains with you, probably for the rest of your life. I think a lot of us can remember bits of poetry that we learned when we were very young. So it's something that lives with you forever.
What bits of Shakespeare have you memorized "by heart"? I can do some rote bits of Dream or Macbeth or even the dreaded balcony scene, but other than as a "go to" bit of text when I need it, there's no love for those passages. I know Sonnet 18 by heart because for years I sang it to my children at bedtime. I know Sonnet 17 by heart because I recited it to my wife during our first wedding dance. I'll admit that I don't know "by heart" many longer passages, just some turns of phrase here and there that truly resonate just right where you need them.