The Arkenstone is definitely not a Silmaril for the following reasons (not an exhaustive list):
- The Hobbit explicitly states that there was only one Arkenstone but there were three Silmarils.
- The Silmarils glow with mingled gold/silver light, the Arkenstone glows with white light.
- The Silmarils are in their permanent homes (sky/earth/water) until the end of the world, the Arkenstone was found before the end of the world.
- The Silmarils will burn mortal flesh, the Arkenstone doesn't.
- The Arkenstone was "cut and fashioned by the Dwarves", the Silmarils were effectively indestructible: "like the crystal of diamonds it appeared, and yet was more strong than adamant, so that no violence could mar it or break it within the Kingdom of Arda".
Ultimately, while it can be argued that Tolkien did derive inspiration from the Silmarils when inventing the Arkenstone (in particular see his use of Eorclanstanas to describe the Silmarils in the Old English versions of texts given in HoME4), inspiration is all that it was and the evidence is just too strong against it being one.
Everything that Tolkien ever wrote about the Arkenstone (aside from incidental mentions) is collected in the text of the Hobbit, so it's worth quoting all of the descriptive passages.
But fairest of all was the great white gem, which the dwarves had found beneath the roots of the Mountain, the Heart of the Mountain, the Arkenstone of Thrain. "The Arkenstone! The Arkenstone!" murmured Thorin in the dark, half dreaming with his chin upon his knees. "It was like a globe with a thousand facets; it shone like silver in the firelight, like water in the sun, like snow under the stars, like rain upon the Moon!"
This is just a descriptive passage and offers no clues to it's nature beyond it being a "great white gem".
It was the Arkenstone, the Heart of the Mountain. So Bilbo guessed from Thorin’s description; but indeed there could not be two such gems, even in so marvellous a hoard, even in all the world. Ever as he climbed, the same white gleam had shone before him and drawn his feet towards it. Slowly it grew to a little globe of pallid light. Now as he came near, it was tinged with a flickering sparkle of many colours at the surface, reflected and splintered from the wavering light of his torch. At last he looked down upon it, and he caught his breath. The great jewel shone before his feet of its own inner light, and yet, cut and fashioned by the dwarves, who had dug it from the heart of the mountain long ago, it took all light that fell upon it and changed it into ten thousand sparks of white radiance shot with glints of the rainbow.
This is the main passage that is probably responsible for much of the "Arkenstone = Silmaril" speculation, yet it fails on many of the points I list above.
The Elvenking himself, whose eyes were used to things of wonder and beauty, stood up in amazement. Even Bard gazed marvelling at it in silence. It was as if a globe had been filled with moonlight and hung before them in a net woven of the glint of frosty stars.
And this is a relatively minor passage that offers nothing much new.
In the end that's all we have to work with. There's nothing else in any of the books, nothing in his Letters.
If it's unsatisfactory to say "it's just a white gem" there is still one candidate left over. In the Silmarillion we learn about the Noldors' discovery of gems, and in particular Feanor's making of artificial gems:
...he it was who, first of the Noldor, discovered how gems greater and brighter than those of the earth might be made with skill. The first gems that Feanor made were white and colourless, but being set under starlight they would blaze with blue and silver fires brighter than Helluin...
If we really must seek an explanation for what the Arkenstone is in Tolkien's writings, then such a gem may provide this explanation, and it need not even be one made by Feanor or any of the Noldor: a gem "made with skill" that is "greater and brighter than those of the earth" could just as well have been made by the Dwarves (see "cut and fashioned by the Dwarves"), the children of Aule.
Update - 10th March 2015
In History of Middle-earth 11, commentary to the Grey Annals paragraph 22, Christopher Tolkien notes the following from a earlier version of the text ("GA 1" is the earlier version, "them"/"they" is the Dwarves, "Lorien" is the region in Valinor, not Middle-earth, and "Enfeng" is an earlier name for the Longbeards):
The conclusion of this paragraph is wholly different in GA 1:
For Melian taught them much wisdom (which also they were eager to get), and she gave to them also the great jewel which alone she had brought out of Valinor, work of Feanor, [struck out but then ticked as if to stand: for he gave many such to the folk of Lorien.] A white gem it was that gathered the starlight and sent it forth in blue fires; and the Enfeng prized it above a mountain of wealth.
Christopher Tolkien then goes on to note that this idea was subsequently rejected owing to chronological problems: Melian had left Valinor more than 100 Valian Years before Feanor was born, and so could not have taken one of his jewels out of Valinor. In the following version it was substituted for the story of the pearl Nimphelos.
However the choice of words here is hardly accidental, and this, then, is a candidate for the Arkenstone (accepting the difference in the colour of the light).
Christopher Tolkien does not touch on this possibility in his commentary, however.