A Put sets a contract that gives you the ability to sell shares at a specific price (strike price) in a specific time frame. When the stock goes below the strike price you "buy" shares at the lower price and "sell" them at the higher price. That difference minus the price of the contract that is your profit. If the share price doesn't go below the strike price or you fail to exercise the option you only lose the cost of the contract.

With a Short you pay a fee to borrow shares that someone else owns. This gives you the ability to actively sell the shares in the market. Then you buy back the shares from the market to return to the shareholder. If you buy back the shares at a price lower than what you sold the shares for that difference minus the fees is your profit. However with short selling if the price goes up you have to buy back at a higher price then you lose money. Since stocks can go up infinitely you can theoretically lose an infinite amount of money.

What makes short selling so insidious is that since shorts actively trade in the market they can manipulate it. If you or your confederates have enough borrowed shares all the shares can be dumped at once causing an artificial deflation of the stock price. Done well enough it causes nervous investors not to buy in and can destabilize the share price for a long time.