FULLTEXT searches are absolutely going to be faster, as kibibu noted in the comments above.
However:
mysql> select COUNT(ID) FROM table WHERE INSTR(Name,'search') > 0;
+-----------+
| COUNT(ID) |
+-----------+
| 40735 |
+-----------+
1 row in set (5.54 sec)
mysql> select COUNT(ID) FROM table WHERE Name LIKE '%search%';
+-----------+
| COUNT(ID) |
+-----------+
| 40735 |
+-----------+
1 row in set (5.54 sec)
In my tests, they perform exactly the same. They are both case-insensitive, and generally they perform full-table scans, a general no-no when dealing with high-performance MySQL.
Unless you are doing a prefix search on an indexed column:
mysql> select COUNT(ID) FROM table WHERE Name LIKE 'search%';
+-----------+
| COUNT(ID) |
+-----------+
| 7 |
+-----------+
1 row in set (3.88 sec)
In which case, the LIKE with only a suffix wildcard is much faster.
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