I wrote about the confusion between these two words three years ago (here). From the evidence we appear to be losing the word ‘sliver’ completely. I recently offered a slice of cake to a very well-educated person who wanted to accept “just a slither, please”. And three separate examples from newspapers in otherwise well-written and spelt articles:
1. “At most, it should represent a slither of a broad investment portfolio” (recommending the purchase of a company’s shares)
2. “A slice of 1840 fruitcake for Queen Victoria and Prince Albert’s wedding sold for £1500, while a slither for the marriage of Queen Elizabeth and Prince Philip in 1947 …” (ignore the fact that she was still Princess Elizabeth in 1947)
3. (after suffering a cold) “you wake in the morning, and the nostrils are crusted and a slither of air has entered, light at the end of the tunnel.”
None of these three satisfies the definition of the OED’s “something smooth and slippery; a smoothly sliding mass”.
In my usage, sliver is almost invariably a noun (the most recent example given in the OED for its use as a verb is from Rudyard Kipling in 1896), while slither is almost always a verb, although I can live with a slither of snakes.