But getting down to individual colors is tricky. There aren't many hues that are found only in one palette but not in the other. Both have pinks, yellows, greens, blues, purples... The differences are mainly in the temperature of the hues - cool or warm? - and it can be hard to convey those differences in a blog.
I will focus on just two colors, two that are inarguably distinct in hue - one from each palette.
First is Bright Spring's very saturated adobe orange. Think rust or pumpkin, but clear, not dull.
Here's Rose McGowan in such a color:
The colors that most easily distinguish Bright Winter from Bright Spring, in my opinion, are its periwinkles. Periwinkle is purple-blue. Bright Winter's are like this:
Now, Bright Spring has several colors you might call purple.
But none of them are purple-blue.
Here's Courtney Cox in a similar color:
For more help figuring out your season, see here. Good luck!