BLAST developed in 1990 has been the most popular nucleotide and protein search program. It is another heuristic implementation of Smith-Waterman based on seeding and extension approach.
The logic is fairly easy to understand, if you followed the logic of Smith Waterman in the last section. As we explained, Smith Waterman algorithm matches a query sequence with a reference by first finding a matching base and then continually extending it based on improved scores for incorporating additional bases using a dynamic programming approach.
BLAST makes use of the observation that most realistic matches constitute of multiple perfectly matching bases clustered together.
Furthermore, BLAST builds an index.
Difference with FASTA -
http://www.bio.net/bionet/mm/bio-soft/1993-November/006173.html