One important point in many of Lovecraft's stories (as opposed to adaptations in later media) is that it isn't often simply seeing a horrific creature that breaks someone's mind, it's the 'revelation' or 'moment of insight' as to what that thing's existence means.
The sciences, each straining in its own direction, have hitherto harmed us little; but some day the piecing together of dissociated knowledge will open up such terrifying vistas of reality, and of our frightful position therein, that we shall either go mad from the revelation or flee from the light into the peace and safety of a new dark age.
-- The Call of Cthulhu
This quote is especially telling, IMO:
Johansen, thank God, did not know quite all, even though he saw the city and the Thing
IE, he would have been worse off mentally if he knew what Cthulhu represented; the horror is not simply the visual appearance of the creature.
In the absence of that, people can "get used to" even the weirdest things - The Shadow out of Time for example; being mind-switched into a Yithian body is originally utterly horrifying, but it doesn't lead to total insanity and people seem to eventually adapt.
The Elder Things, seen as dead specimens, aren't mentally dangerous because they are just considered to be weird prehistoric life forms - you expect odd shapes among fossil sea creatures. (They are originally assumed to be "radiates", which I think is an old term for echinoderms; IE, they aren't at first realized to be extraterrestrial or intelligent.) They don't present a threat to the "worldview" or mental stability of the scientists observing them until the real picture emerges.
The sailors who die from seeing Cthulhu may be a different case, however. Cthulhu - while above the sea - projects telepathic signal that can be received by humans (influencing the dreams of artistic/sensitive/etc. types, and causing increases in violence, thousands of miles away). The killing fear here might not be a purely "internal" reaction.
What Danforth sees at the very end of Mountains of Madness is implied to be something extra-dimensional, far stranger than an Elder Thing or even a Shoggoth...
He has on rare occasions whispered disjointed and irresponsible things about "The black pit," "the carven rim," "the proto-Shoggoths," "the windowless solids with five dimensions," "the nameless cylinder," "the elder Pharos," "Yog-Sothoth," "the primal white jelly," "the color out of space," "the wings," "the eyes in darkness," "the moon-ladder," "the original, the eternal, the undying," and other bizarre conceptions
-At the Mountains of Madness