As I said in my other comment, I had to do a
lot more memorization to learn to recognize typical patterns (like chords and scales) in traditional notation, while integer notation ... well, for starters I just about came up with it myself, before I learned it's (of course) an established way of notation in music theory. Second I just wrote some code to play around with the numbers, basically for myself to try and make sense of the traditional system, because it seemed (to me) easier to convert the bunch to numbers and see what chords come out. What happened is that, really through no effort on my own, I started recognizing the numerical patterns for what they mean in musical theory.
It's really that much more intuitive. Give somebody CEGC' and GBD'G' or give them {0,4,7,12} and {7,11,14,19}. And as a bonus, if you can read clock, you already know how to do modulo 12 arithmetic! 14:00 is 2pm, 19:00 is 7pm, therefore that last one can also be written as {7,11,2,7}.
I don't want to do away with traditional notation, it has its advantages. But I do think that integer notation can be used very effectively and should be used more, when teaching the fundamentals of music theory.