Demons don't have a culture; they function as individuals, much like a shiver of Sharks (yes, that's the real term). Additionally, they are intrinsically supernatural beings, unlike the Goblins, whom are merely normal living things. As such, the Demons are freed from a lot of evolutionary or functional problems.
They presumably have no desires other than intellectual ones- no hunger, thirst, exhaustion, or even motivation for Tentacle Demons1- at least not directly. These intellectual desires- ambition, fulfillment, an extremely difficult to quantify "social" desire encompassing dislike or appreciation for various people, and possibly a unique desire "to be like my alignment".
That last one represents the odd decision making process that would lead a super-entity such as a Demon or God (for now we really must include DF's Gods and any avatars of them in this discussion for the sake of completeness) to be "Bad for Badness' sake", doing things that don't make sense for any reason other than that it's what that alignment would do. Unlike mere mortal morality and it's required pragmatism, this is the sense of how, for instance, some specific good deity will forever trust an evil one, not fighting against him or destroying him altogether, for the sake of giving him the chance to decide to come to her side willingly, even as all the time the evil one is seeking to harm her followers or lead them astray.
As such, Goblin Demons are not motivated by the quest for the results of power, such as wealth and personal comfort. These things are below them, and surely they could find simpler ways to get them if they desired them. Instead, they have emotional, intellectual reasons for wishing to lead the goblins or otherwise meddling in the medium of mortals, a few possibilities below:
-Some abstract source of power- the closest they come to having physical needs- is provided by the emotions of souls at war. Perhaps with the whole of the world worshiping them, the Goblin Demons become full-on dieties.
-They enjoy the abstract tactical concerns of warfare, i.e., they are toying with the lives of millions as... a game. Perhaps once they conquer the world, they will reset it with a more challenging set of opponents. Perhaps in this game, I'll control the Kobolds and we'll start in a haunted desert, they say to themselves.
-They admire or at least have a form of affection for Goblins and truly believe that they themselves make good leaders to bring the goblin race to glory
-A wish to do something constructive- in a sense- by shaping the world as they see fit. The world will be formed to their will.
-Dislike of Humans or whatever-else-there-be; destroying them and making them suffer is the goal, not anything practical. Perhaps tormenting them is so much of a goal that the Demon takes great pains in not annihilating them so as to provide playthings for eternity.
While this makes the DF world sound dark and terrible, let's not forget the presence of Gods who must have similar, opposite motivations, and Heroes fighting against the Demons personally.
Allow me to make one brief comment about the "Moral Relativity" thing as well; this is not that. This is Functional Morality, though I admit I just made up the term, that is that, regardless of what the origin of moral behavior is, (though I personally believe that it evolved,) it has function and use in a society, and cultures with a moral code are stronger and more successful than those without them. What this means for apparently amoral cultures, such as those of the goblins, is that they must have some kind of structure that allows them to function as well as moral ones. The presence of Demons is something I did not consider thoroughly but it seems that they are definitely, like I briefly mentioned, essential to supporting a goblin culture- and probably to maintaining it as it is.
1A footnote about the Tentacle Demons; if they would have no intrinsic physical motivation, perhaps their behavior stems from an intellectual desire to enact a specific form of torment.
Thanks everyone for the insightful comments. (Holy Batcave, I got like 8 new responses typing this...)