Usually, when searching with a normal index, you can search only in a single field, e.g. "find all cities that begin with A" or something like that.
Fulltext index allows you to search across multiple columns, e.g. search at once in street, city, province, etc. That might be an advantage if you want to do something like a Google-style search - just punch in a search term and find all rows that have that search term anywhere in any of the varchar columns.
Additionally, with a regular search, you are fairly limited in what you can do - you can search for an exact match or just LIKE - that's about it.
With fulltext index, you can search for word forms (ran, run, etc.) and also for similar words by specifying your own thesaurus. You can search based on several languages if that's an issue. You can search for entries that have two or more terms that are "NEAR" one another.
Marc