You are asking two different questions here, one in the title ("should I read the Silmarillion before or after LotR?") and one in the text ("what would be the upside to reading the Silmarillion before LotR?"). You have gotten very good answers to the first question, although there are some factual inaccuracies (e.g. if we're going for time-order, it turns out Tolkien started the Silmarillion before starting the Lord of the Rings trilogy) -- but the second question seems quite wide-open.
So let's talk about what this is. The Silmarillion is a tale of gods and elves and men. It begins by explaining that there are three levels of godhood, which one might call in English God Supreme, god, and demigod: these are called by the Elves Eru Ilúvatar, Vala, and Maia. As in Judeo-Christian backstory, the creator category is a singleton -- there is one being inhabiting this category, one True God who created our world by creating the gods and demigods to sing in a sort of harmony. In this story one particular god was better than all of the other gods and demigods, and desired a sort of individuality from the divine which is ultimately in many ways a betrayal of that God Supreme and results in all of the evil, chaos, and discord in the world; he sang discordant notes into the harmony. It is clear that this is a part of the God Supreme's intent and plan, but it still is the source of great strife for us here. Other gods, however, during the story are able to work together to imprison and then exile this god from the world that he helped to create, so that his influences are more subtle and lingering than actively destructive. This sets up the Lord of the Rings books as a discussion of how two of those lingering threads of the betrayer's chaos -- the demigods Saruman and Sauron, who became minions of the betrayer -- were ultimately defeated by the cooperation of men and elves and sentient trees and the demigod who you know as Gandalf.
As people have suggested, this kind of puts the whole epic narrative into a sort of perspective that kind of takes a lot of its punch away. But in some ways it still has a lot of force: this is a critical moment where despite being defeated in the absolute largest battle, the betrayer-god yet has the ability to win the entire war. If this is a game of chess, he has two pawns which are about to make it to the back row and get queened. For some reason the other gods don't seem to see this (or don't realize how serious the threat is or something) and the fate of this cosmic struggle depends only on these poor, ordinary individuals.
I think part of the problem is that therefore, to first approximation you mostly get a ton of backstory, which is not necessary to understand the greater story but realy does help in flavoring it. Like, if you read the Silmarillion first, you get to feel just how impressive Elrond is when later in Fellowship he off-hand says "I remember well the splendour of their banners. It recalled to me the glory of the Elder Days and the hosts of Beleriand, so many great princes and captains were assembled." And you realize that this isn't just someone named after the Elrond of the Silmarillion, this is the same guy, having been around for thousands of years. Like, without that backstory you could maybe have reasonably thought that the "Elder Days" were a few hundred years back, but no: this elf is truly ancient.
You also get a sort of conspiratorial perspective on the rest of the narrative. For example, there is a character who you have not met yet, named Tom Bombadil. He is a lot of peoples' least-favorite character because he appears in Fellowship as a sort of deus-ex-machina saving the hobbits from an early before-they-are-ready encounter with the forces of evil. When you return to these epic books to reread them things only get more confusing, "wait, he did what with the One Ring?!" But once you have read the Silmarillion, you can kind of approach this character with a new perspective, "okay so this is clearly either a God or Demigod, is it maybe one of the ones from the Silmarillion?"
(I should note that there was some ambiguous dialogue which caused J.R.R. Tolkien to write a letter specifically saying that Tom is not Eru Ilúvatar: Frodo asks "Who is Tom Bombadil?" and his wife is not able to really comprehend that question and answers unhelpfully, "He is", but this has an uncomfortable Judeo-Christian resonance as God says to Moses, 'Tell your people that "I AM" has sent you.' So Tolkien shut that fan theory down. But all of the Valar and Maiar are still open for interesting fan theories, and Tolkien is also on record as having specifically left Tom in during editing to leave this very question open, so as to create an enigmatic feeling in the world of his creation. So fan theories are very much this character's point.)
Those sorts of things are what you can expect to get out of reading The Silmarillion before you read The Lord of the Rings. The scope of what's going on becomes vastly more epic, albeit that a lot of the backstory has already passed; a lot of the people who you thought might be normal humans turn out to be deities of various forms; some people turn out to be way, way older than anything you've thought of before; and finally, you may get a better handle on what the heck is going on with Gondor and its Stewards and Aragorn and the Sword that was Broken, because Isildur will be a somewhat more familiar name to you, maybe.