Solution
Place this formula in cell C2:
=ARRAYFORMULA(SORT(UNIQUE(IF(ISNA(MATCH(A2:A,B2:B,0)),A2:A,""))))
What you want is the set difference. This formula will give you all the unique values of column A that do not appear in column B.
Details
MATCH(search_criterion, lookup_array, [match_type]) checks if an element is found in the given array. In this case it checks the element from column A in the entire array of B. If it is not found it returns NA.
IF(ISNA(...), A2:A, "") returns the contents of cell from column A if it was not matched, or an empty string if it was.
SORT(UNIQUE(...)) is optional, but will filter the results in alphabetical order and remove any duplicates which is probably what you want.
Finally ARRAYFORMULA returns the result as an array so that it populates the entire C column.
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