Apple - What directory comparison tools can I use on OS X?

FileMerge (free), shipped with Xcode, offers a directory view.
A command line version is available through the Terminal application opendiff.

Here's how you can compare two directories with FileMerge:

1. ⌘+space, type in "FileMerge" and open it.
2. Click the "left" button and choose the folder you would like to move items FROM. (The "old" folder)
3. Click the "right" button and choose the folder you would like to move items TO. ("new" folder) and click "Compare" button
4. In the right panel, choose to exclude: "identical" and "Changed right". This way you will only see files which are missing in the "new" folder and ignore files your may have added in the "new" folder.
5. Move files manually in Finder or let FileMerge do it, by choosing an option in the "Merge" dropdown in the right panel.

Built-in utility for macOS and Linux

If you don't mind using the terminal, the diff command can compare directories.
This utility is also available in most Linux distributions.

diff -rq directory1/ directory2/

-r indicates recurse through subdirectories, and -q gives brief output (i.e. don't show the actual diffs, just note what files/dirs are different).

Other useful options are:
-s report identical files,
-i ignore case in file contents,
--ignore-file-name-case ignore case when comparing file names.

If you want to avoid warnings (mostly usefulness warnings) about differences in the .DS_Store files, then use:

diff -rq directory1/ directory2/ | grep -v .DS_Store

Diffmerge should meet all your requirements.