Stash is great. We use it at work and it's quite simple yet powerful.

You can set up branch permissions, require at least one approval before merging pull requests, one Bamboo build to pass etc.

Since JIRA 6.1 (I think) you can branch from within JIRA, if it's wired up to Stash. (Although this might be any Git repository.)

Yap, branching from within Jira 6.1 is great. With the OnDemand suite this all just works flawlessly too!

One of the differences is an insane difference in price. According to their two respective pricing pages a 500 user installation would be $12,000 with stash and $125,000 with GitHub::Fi.

I believe volume licenses apply after 100 users, but even so there is a rather large price difference.

If it's really like in this video: https://www.atlassian.com/software-development?utm_source=ji..., then it should be pretty awesome

There is GitHub integration in JIRA. Not as much as with Stash, but it's ok.