I was wondering this same thing but I haven't ordered the White Birch yet, only the Vanilla Birch. Based on the scent descriptions on their websites, I'd have to say these are probably vastly different scents.
White Birch/CS:
Our White Birch fragrance oil adds a new layer of sophistication to traditional pine scents with vivid notes of eucalyptus, cypress, and tonka bean, creating a unique aromatic crispness. A fabulous choice for winter candle or soap lines, this scent also has year-round appeal as a luxury evergreen fragrance.
This fragrance is infused with natural essential oils, including fir needle, cedarwood, eucalyptus, patchouli, and cedar leaf.
Vanilla Birch/WSP:
Quince nectar, mandarin zest and pear brighten a seductive base of sandalwood, birchwood and vanilla.
She may not remember what the VB smelled like so maybe try the switch?