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Should I use Blast or Fasta?

Blast is a program developed at NCBI which searches with a query sequence against a database. Blast looks for regions of local similarity -- i.e. it searches for regions of the same length between the query sequence and the database sequence. It does not insert gaps to improve the quality of the match. Blast calculates a 'quality-of-match' value using a scoring matrix, and the output will have all matches which are above a cutoff value. Since Blast looks for regions of similarity, it can find that two different regions of the same database sequence are similar to a query sequence.

Fasta uses the Pearson-Lippman algorithm, and looks for "word" matches (an optimal word size is 2 for proteins and 4-6 for nucleic acids) between the query and the database sequence. It finds the region in the database sequence that has the most word matches, and then looks for other nearby regions that match (i.e. it inserts gaps into the sequences if that will improve the match). It keeps track of the highest-matching sequences, and will report only the top matching sequences.

Which is more appropriate for your purpose? Note the following:

Because of its speed and simplicity, a Blast search is probably your best first bet. If one of the above points is important to you, or you want more details, try Fasta.

Additional links:

More on Blast and Fasta at U. Oxford.
Database Similarity Searching using Blast and Fasta -- an article by Bruno Gaeta with details about the algorithms and choices.
Interpreting FastA output from the University of Minnesota.
The Blast Help Manual from NCBI.
Searching databases for sequences similar to a sequence of interest from NYU.