Visually, where JPEG produces ringing artifacts, JPEG2000 produces blurriness. JPEG artifacts look "sparkly" and are often a reasonable replacement for the original texture, JPEG2000 artifacts look offensively dull (my opinion). I think, in theoretical speak, it's called "JPEG better preserves high frequency energy". You should be able to find side by side comparisons using your favorite image search engine.
This is the result of actual experiments with human subjects judging image quality.
I haven't really heard claims that JPEG2000 is better, actually. I remember people saying that it was "supposed" to be better, but not by people who dug into it and made comparisons with human eyeballs.
There were some images that would appear visibly better with JPEG2000, like photographs with those beautiful fields of defocused color. JPEG2000 captured those fields of color with all the smoothness they originally had.
Subjectively I think JPEG2000 is better across the different images, at least at this "medium" size preset.