What do mango butter, panthenol and vanilla extract each do in this cream?

The three named components cover different aspects of body-skin comfort:

• Mango butter is a plant butter pressed from the mango seed. Its fatty acids help reduce water loss from the skin's surface and reinforce the hydrolipid barrier — the thin film that keeps moisture in and everyday irritants out. On dry, rough patches this is what produces the softer, more velvety feel and helps skin look less dull.
• Panthenol (provitamin B5) works from the hydration side: it attracts and binds water, so skin feels more flexible and less tight. It is a common choice in moisturisers aimed at the dryness people notice after showering, or from wind and indoor heating in winter.
• Vanilla extract contributes a conditioning effect and is the source of the warm, sweet scent, which is worth knowing if you prefer unscented body care or are sensitive to gourmand fragrances.

The logic of the pairing is that panthenol draws water into the skin while mango butter helps keep it there — a humectant plus an occlusive-emollient combination that is standard in well-constructed moisturisers. Neither ingredient is a treatment: if you have a diagnosed skin condition rather than ordinary dryness, discuss suitable care with a specialist.