The choice between platinum and gold for jewelry depends on personal preference, intended use, and current price of metals.

Platinum

Platinum is a rare and dense metal that is more expensive than gold. It is hypoallergenic, does not tarnish or corrode, and is ideal for people with sensitive skin. Platinum is also heavier and denser than gold, making it a good choice for those who prefer heavier, more substantial jewelry. The amount of platinum that is mined each year is about 10% of the total amount of gold mined.
When used for jewelry, 95% of platinum is used along with 5% alloy. Platinum is the most expensive of metals due to its rarity and density, however it is important to consider the current price of gold. Currently as this blog is being written, the cost of gold is so high that the difference in cost between platinum and gold is very nominal.
It is important to consider that platinum takes on a "patina" over time, which is caused by many tiny little scratches in the metal and gives it a more satin look. This patina, however, is commonly valued by jewelry collectors due to its authenticity.
Platinum is the most expensive of metals due to it's rarity and density, however it is important to consider the current price of gold. Currently as this blog is being written, the cost of gold is so high that the difference in cost between platinum and gold is very nominal.
Platinum is always recommended for any wearer that has or suspects that they have a sensitivity to any metal alloys. Read more below.
 

Gold

Gold, on the other hand, is a soft metal that is widely used in jewelry due to its beauty and affordability. It is available in various karats, with 24-karat being the purest form. 14-karat (56.5% gold & 41.5% alloys) and 18-karat gold (75% gold & 25% alloys) are the most popular options for jewelry, as they are a good balance of affordability and durability.
Most people do not realize that white gold is plated with rhodium, which is a very expensive alloy. This is what keeps the color (because gold is naturally yellow) and keeps it shiny. However for this reason alone, white gold needs more upkeep because pieces need to be replaced every few years. Often white gold may be less expensive upfront, but end up overall costing more in upkeep in the long run. Gold tends to keep it's shiny luster longer than platinum.
One of the more common skin allergies is nickel, which is most often used as an alloy in 10k, 14k, and 18k gold (to different degrees), which means that it can cause skin reactions for anyone with a nickel sensitivity. Other alloys commonly used with gold are copper, silver, iron, and zinc. If you have reacted to any type of jewelry or metal in the past, it is not suggested that you use gold to have jewelry made, as there may be trace amounts of various metals used for alloy.
Ultimately, the choice between platinum and gold depends on your personal style, budget, and intended use.