A release clause is a figure written into a player's contract at which their club must allow talks with a buyer, or must accept a bid outright. A buy-out clause is the stricter Spanish version: the player buys out the rest of their own contract, and the club cannot refuse.
The two get used interchangeably in reporting, but they behave differently. A release clause is a term between the club and the player, and its exact effect depends on the wording. Some only oblige the club to open negotiations, so a bid at the clause figure does not always end in a transfer.
A buy-out clause comes from Spanish employment law, where every professional contract has to carry one. The buying side deposits the sum with the league on the player's behalf, the contract is terminated unilaterally, and the selling club has no say. That is why Spanish clubs set the figures deliberately high.
Clauses are often conditional in ways that never make the headline. They can apply only to clubs abroad, only in a particular window, only above a certain league position, or lapse entirely after a set date. A player and their agent will push for a low, wide clause; the club will push for a high, narrow one.