Do we know anything about the difference in the concept between those two cases?
It's not because it's metal:
Telepathic Resistance/Immunity: The helmet protects its wearer from all but the strongest or most unexpected of telepathic attacks. This is achieved due to technology wired into the helmet itself.
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:
He reshapes the metal at the molecular level so it has the anti-psionic abilities necessary to keep his mind from being or affected by telepathic abilities.
We have watched him shape his helmet, even when it is lost or damaged out of scrap metal if necessary.
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:
(a) technology wired into his helmet (the explanation given in the X-Men film series and several comic plotlines), (b) some physical aspect of his electromagnetic powers that can interfere with telepathy (he once used the Earth's magnetic field to dampen the powers of all telepaths within his reach), (c) latent telepathic powers of his own or (d) sheer force of will (cf. X-Men Vol. 2 #2). The theme of latent telepathic powers has been explored in a number of stories, among them the Secret Wars limited series. In some of his earliest appearances, Magneto was depicted as capable of engaging in astral projection. He has used Cerebro to locate mutants at great distances while leading the New Mutants.[volume & issue needed] He has also, on rare occasions, been shown reading other's dreams, issuing telepathic commands, and probing the minds of others. He has demonstrated the ability to shield his mind, while in intense meditation, so completely that even Emma Frost was not able to read his thoughts, despite being directly in front of him and actively attempting to do so.