I was going to say they're synonymous; checked Wiktionary which calls it 'nonstandard' for sliver, though common in the UK (where I am from and live) blaming 'th fronting'.
I assume that's the name for, ahem, 'that is just anovver word for it'.
I didn't know either, I only really checked because I was curious if they had a completely different etymology and only happened to be spelt^ and used similarly.
(^or if, like spelt and spelled, the same root had just come to be used in two ways for the same.)
So, note to self, slither is not a noun! (Except to mean limestone rubble apparently, but I think I can ignore that.)
In American English, "slither" is more frequently associated with the movement of snakes, specifically. A snake "slithers" by "moving smoothly over a surface with a twisting or oscillating motion." [0]
I think that should be "sliver".