What's actually in each
Platinum 950 (Pt950): 95% platinum + 5% ruthenium or iridium for hardness. Naturally white, no plating needed. White gold (typically 18kt = 750 fine gold): 75% pure gold + 25% alloy of palladium, nickel, copper, and zinc, then rhodium-plated for the bright white colour you see. The plating is 0.5-2 microns thick and wears off over years; white gold rings and watches need re-plating every 5-15 years to maintain their white colour. Platinum doesn't.
What you feel on the wrist
Platinum is heavy: a 40mm platinum watch can weigh 30-50% more than its white-gold equivalent. Some wearers love this 'serious' wrist presence; others find it tiring after a full day. Platinum is softer than white gold: it scratches more easily but the metal isn't lost (it just gets pushed around the surface), so platinum patinas in a way white gold doesn't. White gold scratches less but loses metal when scratched and shows wear differently.
What it costs
Material cost: platinum is currently ~$30-40/gram vs gold ~$60-80/gram, so the raw material is actually cheaper. BUT platinum requires more material per case due to higher density (you need more grams to fill the same case volume), and platinum is harder to machine, requiring slower, more careful manufacturing. Net effect: platinum cases retail roughly 1.5-2x white-gold equivalents at most haute brands. Patek Calatrava in platinum vs white gold is a typical example.
Which to choose
Platinum for prestige: it's the most-expensive case material at most haute brands, signals 'I went above white gold', develops the soft patina collectors love, and on Patek specifically carries a tiny diamond at 6 o'clock as a Patek-only platinum marker. White gold for daily wear: lighter, harder, easier to live with. Both for evening: both look identical at conversational distance; the differences are weight, longevity of finish, and brand-specific positioning. See wiki: white gold vs platinum.