Use git diff.

git diff [<options>] <commit>..​<commit> [--] [<path>…​]

<commit> is a branch name, a commit hash, or a shorthand symbolic reference.

Examples:

git diff abc123..def567

git diff HEAD..origin/master

That will produce the diff between the tips of the two branches. If you'd prefer to find the changes that occurred in the common ancestor since the branch was started off, you can use three dots instead of two:

git diff <commit>...<commit>

To check which files differ, not how the content differs, use --name-only:

git diff --name-only <commit>..​<commit>

Note that in the <commit>..<commit> (two dot) syntax, the dots are optional; the following is synonymous:

git diff commit1 commit2

Git has no real notion of moved/renamed/copied files, however it has heuristics to try and detect those cases and then display them in a nice way.

That said, you can instruct git-diff to use those heuristics to detect renames by passing it the -M flag. With that you can try

git diff -M new_branch -- new-path/file old-path/file

and see if it works in your specific case. Further, git diff -M --stat new_branch will give you an overview of the changes, and also show you what renames Git's heuristics picked up.

Use GitHub’s Compare URL directly
You can manually construct the comparison URL:
https://github.com///compare/...

Replace:
with the GitHub username or org (e.g., octocat)
with the repository name (e.g., Spoon-Knife)

[Edited by @queenofcorgis to add the below]

We have a good example here in the docs for comparing branches and commits as well:

For more information, read Comparing branches in the GitHub Docs.