A penalty shot is awarded during live play as a direct consequence of a specific infraction — most commonly a player being fouled from behind on a clear breakaway with no defender left to beat, or a defending player other than the goaltender closing their hand on the puck inside the crease. It can happen at any point in a game, any number of times if warranted, and once it's over, the game clock and score situation simply continue as normal.
A shootout only happens after a game ends in a tie following a full overtime period — and only in the regular season. Playoff games have no shootouts at all; if a playoff game is still tied after regulation, it moves straight into full sudden-death overtime periods until someone scores, no matter how long that takes. A regular-season shootout is a structured tie-breaking procedure, not a response to any foul that happened during the game.
Despite serving completely different purposes, the two situations share nearly identical mechanics: a single skater, a single goaltender, and no other players interfering with the attempt. That visual similarity is exactly why fans often blur the two together, even though one is a penalty for something that just happened and the other is a scheduled procedure for breaking a tie.
The shootout itself has a specific origin worth knowing: it was introduced when the NHL returned from the 2004-05 lockout, specifically to eliminate tie games and give fans a clear winner every night. The penalty shot, by contrast, has been part of the rulebook in some form for decades longer, existing purely to punish a specific kind of foul rather than to resolve a deadlock.
Same format, different job — one is a punishment, the other is a coin flip with skill involved.