The differences between the 2003 DVD and the 2015 Signature Edition are most evident during scenes that originally looked much cooler. The latter release is dimmer, particularly and especially during those night scenes, and has a noticeably warmer (albeit very rich) palette throughout.
Top: 2003 DVD
Bottom: 2015 Regrade
What’s important to keep in mind is that neither version is “right” or “wrong.” The 2003 DVD is exactly what the movie looked like on 35mm, and the go-to for purists. However, Brad Bird himself supervised and approved the Signature Edition transfer, so in the end, it all comes down to which grading you most prefer, and I tend to be split between them myself. While I mostly prefer the original color timing, tons of individual shots— sometimes entire scenes in the Signature Edition look like they were sourced from an (of course, nonexistent) IB Technicolor print.