If the site is not very popular, then simple queries with LIKE would do the job best. Just write a simple search query parser which would allow users to apply AND, OR and wildcards to it.
MySQL FULLTEXT is for MyISAM tables only and has some other limitations. And it has poor performance. Just
an example benchmark. So if you're running a busy site and have a VPS or a Dedicated server, then Sphinx is the best option.
Summary: PHP+LIKE for small sites, Sphinx for big sites and FULLTEXT for intermediate sites with MyISAM tables which have not been filled with much data yet.