Both hedge funds and short take advantage from bearish markets, but what is the difference between them?
Hedge funds are a form of fund, an adjective and noun. Short selling is a concept and action, an adjective and verb. Those are the primary differences.
Hedge funds aren't obligated to take advantage of bearish markets, or even hedge their positions. The title is a misnomer related to a distinction created in bygone days.
Short selling is an action. Where someone borrows an asset and sells it for the total amount the market will buy it for, with the idea that they can use those proceeds to buy an identical asset for a much lower price (to give back to the original owner), and keep the remainder of those proceeds for themselves.
An illustrative example would be if you borrowed your friend's iPhone 7 plus and immediately sold it today for $800, and bought another one next year for $500 and gave it to your friend. You keep the $300.
One is a trade strategy (shorting a security) and one is an investment fund (hedge fund).
It's often said that taking an inverse position to offset the risk of a decline in another position is a hedge strategy (you're hedging your risk). Buying some put contracts on shares you own is a hedging strategy, because it gives you the option to sell your shares at a predetermined price in the event that they lose value, and that's different than taking a short position in a security. Though taking a short position in a security to offset some risk in your portfolio is both a short trade and a hedging strategy.