The two palettes are clearly different. Look at them side by side and there's no doubt. Bright Winter is darker, crisper, bluer; Bright Spring is livelier, cheerier, yellower.
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.
The Bright Spring palette has a few colors like this, and they're nothing like anything in Bright Winter's palette.
Here's Rose McGowan in such a color:
If you are lovely in some version of this warm, deep, but clear orange, you're more likely Bright Spring than Bright Winter.
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.
Some are crayon purple; others are quite magenta-ish, almost red.
But none of them are purple-blue.
Here's Courtney Cox in a similar color:
If you suspect you're a Bright but aren't sure which one, compare these two colors. Make sure both versions are very saturated, not faded or grayed or dull.
For more help figuring out your season, see here. Good luck!