Differences are how they are used. Somethings involved:

кое-что this is like 'a thing or two' => 'He began to understand a thing or two (a certain something or other).' This would use кое-что.

что-нибудь 'anything' These could use что-нибудь => Anything in there? Anything behind the door?

что-либо can act like a place holder, for example in a sample sentence. For example if demonstrating how a verb acts, что-либо is placed for an object in a sample sentence. Reader understands that he puts person/object of interest in place of что-либо, and uses correct declension for the noun actually chosen.

Related examples from book of idioms:

"пройтись по чьему-либо адресу"
"припирать кого-либо к стенке" Just put in person/object with correct declension.

If these are not correct, I'll be happy to be corrected.