Has a limited number of fields to search by such as Any Word Anywhere, Title, Boolean Search, etc. To do any complex search you have to know a special syntax (+alien or !migration) to tie the terms together. The Quick Limit allows you to narrow your search even more (Last 5 years, e-Books, Films/Video)
If you know exactly what your are looking for, it can give more precise results.
Is often easier to get results with because it gives you more ways to search by, an easier way to combine Boolean Terms (AND, OR, NOT) using drop downs, and is not as picky about the order the words are entered. There are more fields to search from like ISBN (International Standard Book Number), Publisher, Conference Proceeding, and Coporate Author (like the U.S. Department of Defense)
If you know don't know exacty what your looking for, it can give you more search options.
Use Quick Search if you know what your looking for, it will allow you to be more direct.
Use Advanced Search if you aren't sure what your looking for, it allows the most flexibility.