The Institute and the Brotherhood of Steel, from my perspective, represent "Man as a Fallen Angel" and "Man as a Rising Beast" respectively.
The Institute hates the world, so they want to build a progressive utopia to shield themselves from it. They are afraid of the world, so they hide away in a secret base until they can find out how to impose their ideas and "liberate" the world from the oppression of modern society. They are inherently anti-social, so, if building terminator bots furthers their agenda, they will do it. Far from the peace-loving scientists they appear to be.
They are materialists who want to build a world of limitless prosperity. But, as Julius Evola points out in Fascism Viewed from the Right: "[...] a profound crisis is inevitable at the point when prosperity and comfort will finally become boring. [...] [The youth] notice the absurtity and senselessness of an existence that is socialized, rationalized, materialistic, and dominated by the so-called consumer culture".
The Brotherhood of Steel knows that the world is cold, but they choose to face it head-on. They know that they have to rely on each other. And, to rely on each other, they must be strong, which means destroying those things which make them weak. They don't have time to worry about making peace with the world, they only have time to fight. And they will keep on fighting until the degenerative forces of today are finally rooted out and a stronger society, with a stronger Race, takes it's place.
This is maintained by their "climate of high tension", as Evola would put it. They are constantly at war, so they must constantly improve themselves. Not by running from the world and relying on the artificial, but by facing the realities of the world and gaining their strength from it.