Alhough in many cases the two can be exchanged freely, closure represents a greater level of finality and significance than closing.
The closure of a store or restaurant indicates an end to its existence as a business entity. Closing might mean the same, but is commonly used simply to mark the end of the daily business cycle.
Closure has several further meanings not represented by closing:
The emotional state of satisfaction that a situation has ended and will no longer cause stress or discomfort.
Something that is able to be closed, though it is normally open. An example is a pocket with a zipper. This usage is less common.
Various meanings in mathematics and other technical fields.