The two really have nothing to do with each other. In particular, #== is the equality operator and #=== has absolutely nothing to with equality. Personally, I find it rather unfortunate that #=== looks so similar to #==, uses the equals sign and is often called the case equality operator, triple equals operator or threequals operator when it really has nothing to do with equality.
I call #=== the case subsumption operator (it’s the best I could come up with, I’m open to suggestions, especially from native English speakers).
The best way to describe a === b is “if I have a drawer labeled a, does it make sense to put b in it?”
So, for example, Module#=== tests whether b.is_a?(a). If you have Integer === 2, does it make sense to put 2 in a box labeled Integer? Yes, it does. What about Integer === 'hello'? Obviously not.
Another example is Regexp#===. It tests for a match. Does it make sense to put 'hello' in a box labeled /el+/? Yes, it does.
For collections such as ranges, Range#=== is defined as a membership test: it makes sense to put an element in a box labeled with a collection if that element is in the collection.
So, that’s what #=== does: it tests whether the argument can be subsumed under the receiver.
What does that have to with case expressions? Simple:
case foo
when bar
baz
end
is the same as
if bar === foo
baz
end