Update 2017 (2 years later)

Jaybeecave mentions in the comments the tool diff2html.xyz, a diff parser and pretty html generator.

The git diff format is inspired by the diff -p unix command.
(with -p being for --show-c-function: Show which C function each change is in.)

As I explain in "Where does the excerpt in the git diff hunk header come from?", that feature ("Show which C function") has evolved to take into account other language.

This is similar to what you see in the patch field of the JSON answer when you compare two commits with the GitHub API.
That feature was introduced in December 2012

Simply use the same resource URL and send either application/vnd.github.diff or application/vnd.github.patch in the Accept header:

curl -H "Accept: application/vnd.github.diff" https://api.github.com/repos/pengwynn/dotfiles/commits/aee60a4cd56fb4c6a50e60f17096fc40c0d4d72c

Result:

diff --git a/tmux/tmux.conf.symlink b/tmux/tmux.conf.symlink
index 1f599cb..abaf625 100755
--- a/tmux/tmux.conf.symlink
+++ b/tmux/tmux.conf.symlink
@@ -111,6 +111,7 @@ set-option -g base-index 1
 ## enable mouse
 set-option -g mouse-select-pane on
 set-option -g mouse-select-window on
+set-option -g mouse-resize-pane on
 set-window-option -g mode-keys vi
 set-window-option -g mode-mouse on
 # set-window-option -g monitor-activity off

The format follows the classic diff unified format (also detailed here).
You can see an example in cubicdaiya/node-dtl (an dtl(diff template library) binding for node.js)

For those looking for more clear answer:

const nodegit = require('nodegit');
const repo = await nodegit.Repository.open(repoDirectory);

const from = await repo.getCommit(fromCommitSHA);
const fromTree = await from.getTree();

const to = await repo.getCommit(toCommitSHA);
const toTree = await to.getTree();

const diff = await toTree.diff(fromTree);
const patches = await diff.patches();

for (const patch of patches) {
    console.log(patch.newFile().path());
}

Every patch represents a modified file and is an instance of ConvenientPatch. It has two methods oldFile() and newFile() which return an instance of DiffFile representing the file before and after modification.

NodeGit API docs:

• ConvenientPatch: https://www.nodegit.org/api/convenient_patch/
• DiffFile: https://www.nodegit.org/api/#DiffFile
• All API docs: https://www.nodegit.org/api/