What is the connection or relationship between the two John Phillip
Sousa marches, The Thunderer and The Washington Post. It isn't that
they were both written in the same year. My son's band conductor asked
this question and I cannot find the answer in my own online search.

They were both written while he was the leader of the Marine Corps Band?

?In 1880, Sousa was appointed leader of the Marine Corps Band, which
he would serve for 12 years, under five presidents. He now began to
hit his stride with his own marches, turning out such classics as
?Semper Fidelis,? ?The Washington Post,? ?The Thunderer,? and ?The
High School Cadets.?

http://www.sibeliusmusic.com/cgi-bin/show_score.pl?scoreid=31993

You can rule out time signatures, if you haven't already, because one
is in 6/8 and the other in 4/4. Also, I found inconclusive evidence
that they are not in the same march form. The Washington is in
standard Military March form, but the Thunderer is not mentioned as
composed in that form.

There's an outside chance the connection may have something to do with
the recording history of the two marches. According to a review in the
American Record Guide, the Marine Corps Band under Sousa ?was one of
the first organizations to record, and there are two marches recorded
in the 1890s here, one on each disc: Thunderer and Washington Post.?
From Guide to Records: Sousa ,  By: Hecht, Roger, American Record
Guide, Sep/Oct99, Vol. 62, Issue 5. [This information is from a
subscription database, so I can?t give you a live link.]

This information correlates with the information I found on the
commemorative recording ?From fife and drum: Marine Band Recordings,
1890-1988.? The recording information lists Sousa as the director in
recordings of both the Washington Post and the Thunderer, listed as
from 1890 and 1896 (respectively). However, these are not the only
marches to have been recorded this early, although they are the only
two 19th-century recordings to appear on this album.

Also, both marches have been used in movie soundtracks, but they are
not exceptional in this distinction. I did find one movie that used
both marches:

"First Monday in October," 1981, released Paramount Pictures 
http://www.hollywood.com/movies/fulldetail/id/226403 [See music credits]