A sliver is something that you can pull out with a pin and maybe tweezers. Think of it as a piece of metal that breaks free from the tap and inbeds into your hand.

A splinter is a object that's manlier and deeper than an sliver and made of wood. And, In most causes, just easier to just drive it out the other side.

P.S. Real men don't get slivers. Just angry festering wounds that we use our utility knifes to perform a open hand surgery on the site.