There's a few advantages to full text searching.
Indexing:
Something like:
WHERE Foo LIKE '%Bar';
Cannot take advantage of an index. It has to look at every single row, and see if it matches. A fulltext index, however, can. In fact, fulltext indexes can offer a lot more flexibility in terms of the order of matching words, how close those words are together, etc.
Stemming:
A fulltext search can stem words. If you search for run, you can get results for "ran" or "running". Most fulltext engines have stem dictionaries in a variety of languages.
Weighted Results:
A fulltext index can encompass multiple columns. For example, you can search for "peach pie", and the index can include a title, keywords, and a body. Results that match the title can be weighted higher, as more relevant, and can be sorted to show near the top.
Disadvantages:
A fulltext index can potentially be huge, many times larger than a standard B-TREE index. For this reason, many hosted providers who offer database instances disable this feature, or at least charge extra for it. For example, last I checked, Windows Azure did not support fulltext queries.
Fulltext indexes can also be slower to update. If the data changes a lot, there might be some lag updating indexes compared to standard indexes.