Do we know anything about the difference in the concept between those two cases?
It's not because it's metal:
How exactly it's worked has changed over time, as seen in this answer from @FuzzyBoots.
Except in a few types of media, it's not depicted that the helmet's metal is part of the protection. As that other answer shows, it hasn't always been necessary for Magneto to have his helmet at all.
However, if it were metal alone, then essentially any tinfoil hat would protect against telepaths in the Marvel universe, leading to most characters would adopt them.
To add a little confusion to this, the technology embedded in his helmet isn't always a necessity. In this answer from Thaddeus Howze, our resident comic historian, he explains that Magneto alters the properties of the helmet's metal:
His answer suggests that there's actually a chance Magneto could magnetize Wolverine's adamantium in such a way to make Wolverine immune from psionic attacks, depending on the writer's need. (Imagine Wolverine going crazy and Magneto blocking Xavier/Jean from assisting him!)
Thanks to our community members for already-existing answers that help support this one.
In the comics at least it may not be his helmet that protects him. He is often shown as having psychic abilities of his own. Wikipedia has this to say about it: