In The Lord of the Rings, the One Ring has many powers which make it incredibly dangerous to anyone who holds it, or even comes within close proximity to it. It inspires jealousy, possessiveness, mistrust, discord, infighting, treachery, betrayal, murder, and hatred.
In The Silmarillion, the Silmarils appear to share many of these characteristics. They provoked Fëanor and his kin to take an evil oath, dooming themselves and the entirety of Arda - and even Aman, albeit indirectly. The Silmarils are the cause of much of the suffering and bloodshed of the Elder Days, and countless Men, Elves, Dwarves, and others die as a result of the Silmarils' effects on all who see them. Really, the whole story of The Silmarillion is a long, mournful tale of the misery and malice created by the Silmarils.
There are other parallels between the Ring and the Silmarils. Sauron desperately sought the Ring, much like his master Morgoth sought the Silmarils (although for very different reasons). Some part of the power of the Ainur was locked in each of these objects. Both the Ring and the Silmarils were, more or less, cursed, and brought pain and suffering upon those who possessed them.
Almost anyone who looks upon the Ring or the Silmarils is overcome with covetousness, to such a degree that, even if they are otherwise decent people, they can be inspired to commit horrible acts and unprovoked violence. Both the Ring and the Silmarils are physically beautiful, but are the catalysts for unimaginable ugliness. Both the Ring and the Silmarils are responsible for the deaths of many great leaders and virtuous people.
And so on. As I understand it, much of The Silmarillion was written before Tolkien began working on The Lord of the Rings; however, The Silmarillion was published years after he died, and decades after The Lord of the Rings was published. And although the One Ring appears in The Hobbit, this was actually a retcon, and the description of the ring in the first edition of The Hobbit is nothing like the description in The Lord of the Rings; in the first edition of The Hobbit, the ring is simply an invisibility device with no apparent evil characteristics. It isn't particularly remarkable, in fact, and there is no indication of the far more sinister qualities associated with the One Ring in The Lord of the Rings.
This begs the question of whether he borrowed some of his own ideas regarding the Silmarils and attributed them to the One Ring. Is there any evidence to support this notion? Did he ever write anything along these lines (presumably in his letters)?
Note: While the Ring and the Silmarils share many of their less pleasant attributes in common, there are also some very obvious differences between them. The One Ring is inherently evil, because it was created by an extremely evil being for incredibly evil purposes. The Ring is almost a physical embodiment of evil.
The Silmarils, on the other hand, were not evil at all. The Elf who made them, Fëanor, became quite wicked later on (as a direct result of having created the Silmarils), but when he was making them, he was a nice enough person. And whereas the Ring was made by an evil being using evil materials in evil ways for evil reasons, the Silmarils were made from the light of the Great Trees of Valinor, which were themselves created by the purest, most un-evil beings in the universe - the Ainur; the materials from which they were made were from Valinor, and were therefore also not evil.
The Ring was always, inextricably and irrevocably evil, and it was always intended to be evil. Of the Silmarils, quite the opposite is true. It wasn't the objects themselves that were evil, but the greed and conceit which they inspired in their creator. If Fëanor hadn't sworn a wicked oath upon them, they would have remained beautiful jewels, and proof of a remarkable feat of craftsmanship, and nothing more.