I am impressed by people who can memorize music and then perform that music from memory. I have (thankfully) only had to do it a few times in my life. The first time was in a lesson with Julius Baker during my first year at Juilliard. I was playing a Casterede etude for him. He asked me if I could play it from memory, and I had practiced the piece so many times (for many years) that I was able to play it for him without looking at the music (It was a short etude, and I had just played it with the music in front of me.)
I must have been playing that etude by ear and by feel. I wouldn't say that I was playing it by memory. I don't know if I could have done it again. Luckily I didn't have to find out. When my mother studied with Julius Baker in the 1950s, she had to play everything from memory (which she could do: she had absolute pitch and an incredible memory). I'm grateful that Baker softened up a bit by the 1970s, but he was still impressed by people who could play from memory.
When playing scales and arpeggios on the flute without looking at music, I have to take time to think about what notes I might be playing. When playing scales and arpeggios on the viola or the violin, I have to think about what position I might happen to be in, what instrument I am playing, and what string I happen to be on before I could begin to tell you what scale or arpeggio I might be playing.
I do not memorize music well. I have tried. Again and again. I can rattle off songs I learned long ago, but my interpretation and understanding of them hasn't changed since my adolescent brain imprinted them in the "permanent" section of my memory banks. There are theme songs to television shows, songs from musicals and operettas I did in Junior High and High School, the first dozen or so lines of the poem, "Cut" by Sylvia Plath that I recited as part of an "avant garde" band piece we did in high school where everyone had to recite a different poem at the same time, songs I sang with my kids when they were little, much of the Mozart D major Flute Concerto, Syrinx, the Gluck melody from Orpheus, and the Baker set of daily warm-up excerpts. These are things I learned by rote.
I can make it through some of the first movement of the Bach E-major Partita on the violin without the music in front of me, but I always end up modulating to an impossible key before I realize that I have gotten myself off track. I can also make it through the first movement of the Bach G-major Cello Suite once in a while, and occasionally I surprise myself to find that I can play other movements in the Cello Suites without music. But I can't tell you which ones.
When I practice the E-major Partita (in A major on the viola) with music, I get new musical insights every time. And when I practice the G-major Cello Suite with music I learn something new every time. For me having the music in front of me allows me the freedom to group notes in new (for me) ways. Having the music in front of me helps me to really know where I have been, where I am, and where I am going. It gives me a foot hold. It helps me feel at ease playing in front of people. Playing with the music in front of me becomes more about the music than it does about my playing of the music.