GC
All 12 rings that make up the difference between the versions are right here in the beginning, right after the boarding section. Break the iron container and activate the switch. On Dreamcast, a line of 18 rings appears, but on Gamecube this was reduced to 16. Unfortunately it's hard to tell this difference in the pictures because the rings farther away are cluttered together.
GC
This is the difference more people are familiar with. The item capsule next to the wall near the light dash rings is 20 rings on Dreamcast and 10 rings on Gamecube.
Maximum rings for Dreamcast is 482; Maximum rings for Gamecube is 470.
GC
A short time after making the descent down the tall building, you'll come across this chasm. What you're meant to do is homing attack the beetle and then light dash across the line of rings. On Dreamcast, there are 7 rings here, but on Gamecube there are 9.
Maximum rings for Dreamcast is 441; Maximum rings for Gamecube is 443.