Spelunking a bit in the MAFFT source code, I found the --distout option. It will create a ".hat2" file with the distance matrix.

Apologies for updating an old topic, but this page is one of the first results when searching Google for "MAFFT distout". I'd like to share some information on the distout feature that comes directly from Dr. Katoh, author of MAFFT:

The --distout flag is just for my personal use and there is no document. I think it's better and safer for you to use your code to assess the similarity or distance of sequences from a given alignment. However, I think I can explain this option. Please check if it'll be useful for your purpose or not:

The --distout option outputs distances that are used for building a guide tree. Note that the distances are computed before building an MSA, not computed from the MSA. The distances are converted from pairwise scores with this equation,

Distance(i,j) = 1 - Score(i,j) / min( Score(i,i), Score(j,j) )

as explained in our old paper, Katoh et al 2002. Score() can be computed by various ways (without MSA). In Katoh2002, Score(i,j) was the number of 6mers that are shared by sequences i and j. If the --localpair flag is set, then Score(i,j) is the pairwise alignment score between sequences i and j. The pairwise alignment is not always the same as the resulting MSA even if the number of the input sequences is two.