There is no difference, and both are used also in spoken language.The dictionary translates both the terms שום דבר and כלום as "nothing".
How do they differ in meaning/usage?
Hebrew is similar to English as you need to negate כלום and שום דבר, e.g. in אין לי שום דבר. Colloquially one can answer מה קרה by שום דבר, but this is technically inaccurate, a more accurate answer would be לא קרה שום דבר.Additionally, both basically mean "something" but became somehow "nothing":
In English you can either say "I don't have anything.." (anything ~= something) or "I have nothing..", and formulation like "I don't have nothing" is wrong due to the double negation. However a language has its own way to evolve, and the result is that in Hebrew something=nothing..
Haha yeah, It's a common Israeli folk etymology I think. At least it has the slightest merit synchronically I guess, since Modern Hebrew hasThanks. In ulpan class ages ago, the teacher told us that שום דבר (nothing) actually derives from the word for "garlic", and it's not just some coincidence.