Searching is looking for something, reaching for something, trying to find what you think will give you what you want.
Self inquiry is discovering what is already here. If there is still activity it is directed toward the source rather than objects.
We can make self inquiry a search. We can do self inquiry to get something, and then it becomes searching.
But the purity of self inquiry is in the willingness to stop the outward flow of thinking. Thinking is normally about gathering information for survival. By overriding that very natural instinct, and simply telling the truth about what is here, what you are experiencing, you discover yourself.