How does like query work with indexed table
Let me help you to understand how a "standard" index works.

Most databases indexes are just B-Trees (not to confuse with a binary tree). Simply speaking, when you query an indexed column, a binary search will be performed. A binary search performs generally in O(log(n)) and thus you can find individual rows quite fast, even if there are many of them. The database uses B-Trees instead of loading and sorting the table as the indexes don't require as much memory and less disk reads are required.

Now imagine that you try to binary search for a value, but you don't know the exact beginning or ending of the value. A binary search is basically not possible and you have to traverse nearly the whole tree to find every possibility.

Of course, there're cleverer techniques than this. MySQL is not that dumb, it uses a Boyer-Moore algorithm for this problem, but that doesn't mean you don't suffer a performance impact.

A Fulltext search index will of course help. It uses entirely different datastructures (Tries, Suffix-Trees). Reading the manual of MySQL, I also get the impression that fulltext searching is quite easy with it. 

However, on most systems fulltext searches require some maintenance/housekeeping by the admin to keep up good performance. Often times fulltext indexes keep "mapping/index" tables for the tokens of the indexed texts. These tend to fragment which can impact query response time, as the index grows unecessarily larger. So from time to time, they should be defragmented and optimized. You might want to look into that.