Since I want to use it often, I made @jthill's answer into an alias.

Add the alias with

git config --global alias.branch-diff "!f(){ git diff $(git merge-base $1 $2) $2 "${@:3}"; }; f"

• See https://stackoverflow.com/a/3322412/5057078 for git aliases.
• See https://stackoverflow.com/a/3816747/5057078 for the @ syntax.

With syntax git branch-diff <arg1> <arg2> <more args>. The 1st argument represents the master branch, which the branch in the 2nd argument split off at some point. Further arguments are passed to the diff.

Usage:

1. What files did i change in branch?

git branch-diff master other-branch --name-only

The alias expands this into this and executes it:

git diff $(git merge-base master other-branch) other-branch --name-only

2. What are the actually changes (diff) i made to the branch?

git branch-diff master other-branch

The alias expands this into this and executes it:

git diff $(git merge-base master other-branch) other-branch

Bonus:

# get all file names changed
git branch-diff master other-branch --name-only
# get changes of my_file.cpp
git branch-diff master other-branch -- my_file.cpp