Full text search is likely to be quicker since it will benefit from an index of words that it will use to look up the records, whereas using LIKE is going to need to full table scan.
In some cases LIKE will more accurate since LIKE “%The%” AND LIKE “%Matrix” will pick out “The Matrix” but not “Matrix Reloaded” whereas full text search will ignore “The” and return both. That said both would likely have been a better result.