To memorize is to remember. To say "I remember the deeds of the Lord" is to say "I have memorized the deeds of the Lord."
I have - finally - a "smartphone." This is my first one. It has a lot of "memory." In my smartphone's memory bank is the entire Bible, in three versions. I can access the phone's memory bank if I want and read Scripture. This is cool. And good. But unless the words and deeds of the Lord are in my memory bank, in my heart and mind, something is missing.
Memorization is mostly a lost art today. We have become memory-lost people, having deposited our memories in the memory banks of our technology. Our devices get upgraded while our own hearts and minds get depleted. As our phones grow constantly we are becoming the hollow people. How unfortunate, since it's us that need the transformation and not our phones. (Are our phones a bio-extension of our own human being? Ultimately, I think not.)
If you have a PrayerLife I recommend adding the Psalms to your meditation. To meditate is to memorize, by osmosis. Start with Psalm 1. Take a piece of this psalm and say it over and over and over. As you do this a worldview enters your being and you are changed. N.T. Wright writes (I quote this again - thanks V.H.):
"The regular praying and singing of the Psalms is transformative. It changes the way we understand some of the deepest elements of who we are, or rather, who, where, when, and what we are: we are creatures of space, time, and matter, and though we take our normal understandings of these for granted, it is my suggestion that the Psalms will gently but firmly transform our understandings of all of them. They do this in order that we may be changed, transformed, so that we look at the world, one another, and ourselves in a radically different way, which we believe to be God's way." (N.T. Wright, The Case for the Psalms: Why They are Essential, p. 6)