I believe you could use the following, in combination:
$ diff sqlite-file-1.sql sqlite-file-2.sql > sqlite-patch.diff
$ patch -p0 sqlite-file-1.sql sqlite-patch.diff
I hope that works for you. Otherwise, I highly suggest consulting the man-pages:
$ man diff
$ man patch
EDIT: Alright, here's the whole walk-through.
First, dump the databases:
$ sqlite test1.sql .dump > test1.sql.txt
$ sqlite test2.sql .dump > test2.sql.txt
Next, generate a diff file:
$ diff -u test1.sql.txt test2.sql.txt > patch-0.1.diff
And, finally, to apply the patch:
$ patch -p0 test1.sql.txt patch-0.1.diff
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