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.