No. LIKE does not make use of full text indexing. See here.
Computers are pretty darn fast these days but if you're seeing search results faster than you expect it's possible that you simply got back a cached result set because you executed the same query previously. To be sure you're not getting cached results you could use DBCC DROPCLEANBUFFERS. Take a look at this post for some SQL Server cache clearing options.
Excerpt from the linked page:
Comparing LIKE to Full-Text Search
In contrast to full-text search, the LIKE Transact-SQL predicate works on character patterns only. Also, you cannot use the LIKE predicate to query formatted binary data. Furthermore, a LIKE query against a large amount of unstructured text data is much slower than an equivalent full-text query against the same data. A LIKE query against millions of rows of text data can take minutes to return; whereas a full-text query can take only seconds or less against the same data, depending on the number of rows that are returned.