Why does my bottle of rum say it's "80 proof" instead of just saying 40 percent?

Because the liquor industry has jargon just like any other.

Pure grain alcohol (AKA PGA) is considered to be 200 proof.

‘The term originated in the 18th century, when payments to British sailors included rations of rum. To ensure that it had not been watered down, the rum was “proved” by dousing gunpowder in it, then testing to see if it would ignite. If the gunpowder did not burn, the rum contained too much water—and was considered to be “underproof.”

A “proven” sample of rum was defined to be 100 degrees proof. This was later found to occur at 57.15% ethanol.’

—more at Wikipedia

And then later on, the U.S. standardized on the Alcohol By Volume standard, where degrees of proof is twice the alcohol content percentage.

Rob is right. I work in the industry. I have to know the “jargon.”