The GitLens icon will show up in nav bar. Click on it.
Click on compare
Select branches to compare
Now you can see the difference. You can select any file for which you want to see the diff for.
Update: As of November, 2020, Gitlens appears within VSCode's builtin Source Control Panel
I would recommend to use: Git Lens.
Use the Git History Diff plugin for easy side-by-side branch diffing:
Visit the link above and scroll down to the animated GIF image titled Diff Branch. You'll see you can easily pick any branch and do side-by-side comparison with the branch you are on! It is like getting a preview of what you will see in the GitHub Pull Request. For other Git stuff I prefer Visual Studio Code's built-in functionality or Git Lens as others have mentioned.
However, the above plugin is outstanding for doing branch diffing (i.e., for those doing a rebase Git flow and need to preview before a force push up to a GitHub PR).
Now it's available:
Until now it isn't supported, but you can follow the thread for it: GitHub
It's now possible by using the githistory extension.
Here's a small trick though: You can compare the latest commits from each branch and that would be the same as comparing two branches side by side or creating a PR.
Here's how to do that using githistory extension:
If you just want to view the changes to a particular file between the working copy and a particular commit using GitLens, the currently accepted answer can make it difficult to find the file you're interested in if many files have changed between the versions.
Instead, go to the file explorer in the side bar and right click on the file, go to Open Changes > Open Changes with Revision... (or Open Changes with Branch or Tag...).