Do these two words have any common root?
I looked up in the RAE, and didn't find there any connection between these words. According to the RAE, cuchillo comes from Latin "cultellus", and cuchara comes from verb "cuchar".
So it looks like there are no real connections there, maybe it's only this, but I still see that both words share the common part "cuch". And it seems to me that cuchillo also comes from cuchar, with just adding the suffix -illo.
So how come that the Latin word
cultellus transforms itself from knife to something spoon-like?