What is the difference between a sliver and a chip in the meaning of broken piece of something?
A sliver is small, narrow, and most likely sharp enough to penetrate the skin.
A chip is any small piece broken off from something else without reference to shape or sharpness, though dictionary.com seems to think chips are small and slender as well. Wood chips that come from a wood chipper or the wood chips you buy at a garden store are small but rarely narrow.
A chip can also be what it is and what it isn't: a glass or piece of crockery might have a chip, that is, a chip missing, but the missing chip is, obviously, also a chip.