In The Mandalorian they make a big deal of Mando never removing his helmet in front of others, yet characters like Sabine do it casually all of the time. Is this a contradiction, unexplained, or is there an explanation?
There's a couple of things at play here (note that there's a lot of spoilers here)
Mandalorian has two meanings
In a normal world, Mandalorian would refer to people from Mandalore. In the Star Wars universe, everything is fluid. As such, a Mandalorian can also refer to someone who believes in The Mandalorian Way (I'll just call it The Way for simplicity, since that's what they typically call it, although it does have another name below).
In the series, we learn that
Din Djarin (the namesake Mandalorian for the series) is not from Mandalore. He was saved by Mandalorians and became a follower of The Way.
Lots of retcons
There's no way to get around the fact that Disney keeps retconning Mandalorian canon. Current canon is
Mandalorians were a proud warrior culture that waged war against the Jedi (and presumably the Republic). Since [The Empire] dictates terms [after winning the war], Mandalorians were probably forced to disarm and shun lots of their traditional beliefs and customs.
As such, the best explanation here is that Mandalore Mandalorians weren't strict followers of The Way. After the Empire nearly wiped them out, The Way became a religion that anyone could become a follower of. As such, helmet removal became prohibited among its followers. The Mandalorian Episode 1 dialog supports this
KUIIL: You need to drop your rifle.
MANDALORIAN: I'm a Mandalorian. Weapons are part of my religion.
Bo-Katan Kryze of Mandalore more or less confirms all this in Episode 11 (S2E3). She refers to Din Djarin as thus (taken directly from closed caption)
BO-KATAN: And you are a Child of The Watch.
DIN DJARIN: The Watch?
BO-KATAN: Children of the Watch are a cult of religious zealouts that broke away from Mandalorian society. Their goal was to re-establish the ancient way.
They clearly have disdain for one another when Bo-Katan removes her helmet and Din flies away in apparent disgust after he dismisses what she's just told him.
In S02E03 - Chapter 11 - The Heiress, it has been revealed that Mando belongs to a group of Mandalorians called The Watch.
The Children of the Watch, or simply the Watch, was, as recounted by Bo-Katan Kryze, a Mandalorian cult of religious zealots that followed the Way of the Mandalore, which had fallen out of the mainstream Mandalorian society. The Fighting Corps were members of the Children of the Watch but also bore the symbol of Death Watch.
Din Djarin was a Child of the Watch, having once believed that all Mandalorians were like him.
From the Star Wars Fandom article on The Children of the Watch
If you follow the link to the Way of the Mandalore, the following is being stated:
The faith practiced by the Children of the Watch cult,2 the Way of the Mandalore involved protecting fellow Mandalorians but also wearing a helmet at all times.
From the Star Wars Fandom article on The Way of the Mandalore