Why do the NFL's overtime rules allow for a team to potentially lose without their offense ever taking the field? What caused the league to decide on this format for OT and why did they decide to have different rules during the season and in the playoffs.
It seems very different from OT rules in all other American sports leagues...
More details on the rules here
Why do the NFL's overtime rules allow for a team to potentially lose without their offense ever taking the field?
NFL's overtime is a variant of sudden death in that if the first team scores a touchdown, that team wins.
What caused the league to decide on this format for OT?
The 2009 NFC Championship Game. In short, the Saints won with a field goal without the Vikings taking the field. This was the sudden death format before they altered it in response to this game.
On the first possession, if the team in possession of the ball scores a field goal, the other team has an opportunity to score. In other words, NFL overtime is sudden death unless the first possession results in a field goal.
Why did they decide to have different rules during the season and in the playoffs.
There are different rules during the season vs. the playoffs because in the playoffs, a winner needs to be determined. In the regular season, a tie can be achieved.
I'm adding to the accepted answer to respond to comments in the question.
It seems very different from OT rules in all other American sports leagues
Every American sports league is designed to accommodate television viewership, so when OT rules are different from regulation rules, the trade-off is to end the game quickly without seeming overly unfair or unusual.
In NHL, OT is sudden death. Since possession is fluid and earned when the puck drops, this isn't inherently unfair. However, to shorten OT, teams play 3-on-3. This opens up space on the ice to increase scoring, but some think it's too unlike 5-on-5, which disadvantages some teams more than others.
In NBA, scoring is frequent and possession is earned on tip-off, so it's easy to add 5 minutes of regular play and determine a winner. This is the league with the least drama concerning OT rules.
In MLB, scoring is infrequent, but possession is deterministic: each team will get 3 outs every inning. The current method of extra innings is fair, but it leads to some very long baseball games, which is why the league has considered starting extra innings with a runner on second to increase scoring and determine winners faster. The trade-off is to sacrifice some of the "purity" of the sport to make OT more watchable for fans and less tiring for players.
In NFL, the old sudden death rules -- first score wins, period -- ended games quickly, which is good for a sport that often televises games back-to-back-to-back on Sundays. But it was blatantly unfair, with possession determined by a coin flip. The new rules are fairer, but not ideal. Still, the league determined that fans and players are happier with this than if an entire quarter of normal football were played. Other leagues implement shoot-out-style OT, where offenses take turns scoring from a set yard line -- this faces similar criticism to NHL's 3-on-3, where the game is sufficiently changed such that certain teams benefit more in OT than they do playing by the "real rules".
why did they decide to have different rules during the season and in the playoffs
As exchange_knowledge noted, the primary difference is that a playoff requires a winner and loser.
Another important difference is that in the playoffs, since the stakes are higher, the television audience is willing to watch a longer OT. So instead of prioritizing timeliness, leagues with unusual OT rules prioritize fairness.
For example, if the MLB ever implements a runner-on-second in OT, they will not do so in the playoffs. And in the NHL playoffs, teams play 5-on-5 sudden death periods instead of 3-on-3, and the shootout is also removed.
