Output the common lines (similarities) of two text files (the opposite of diff)?

How about using diff, even though you don't want a diff? Try this:

diff --unchanged-group-format='@@ %dn,%df 
  %<' --old-group-format='' --new-group-format='' \
  --changed-group-format='' a.txt b.txt

Here is what I get with your sample data:

$ cat a.txt 
Foo Bar
X
Hello
World
42
$ cat b.txt 
Foo Baz
Hello
World
23
$ diff --unchanged-group-format='@@ %dn,%df
%<' --old-group-format='' --new-group-format='' \
  --changed-group-format='' a.txt b.txt
@@ 2,3
Hello
World

grep -Fxf file1 file2

-F means match plain strings (not regexps), -x means only whole-line matches, -f means take 'patterns' (i.e. lines) from the file named as its argument

comm can be used. man comm for all the options but you'll want to use comm -12 ... to show only lines that exist in both inputs.

As people have pointed out, you need to pass your input through sort first.