Version 2 of the WordPress REST API has been released on WordPress.org as a whole new plugin.
The decision to do this was so that backward compatibility could be maintained on version 1 on a permanent basis. So if you are running version 1 and wish to run version 2, you'll need to install the new plugin.
This is a huge milestone for the plugin, and helps set the stage for a core WordPress proposal. If you've been waiting to use the API in projects until it is more stable, now is a good time to start. While it can still change, Version 2 is much closer to “production ready” in terms of the schema and structure than Version 1 was. The beta tag it carries is because technically it could still break back compat in the future.
The team is having a bit of a catch 22 with the API right now, as was discussed at length in the REST API channel on WordPress Slack. It's hard to propose it for core while they need more people using it. And more people are nervous to use it for real projects until it has an official date for core.
There really needs to be an official backing for the API, even if it's not given a firm core WordPress version for inclusion (sometime in 2015 was previously the modus operandi). The team would like to get a firm date, or at least get some major plugins (like WooCommerce or Jetpack) to adopt it.
Without significant testing and many more eyeballs, some fear the need for future re-working that may be needed, and if that's combined with promises of back compat now, it could get very tricky.
What the API really needs is attention from people that will build apps to consume it. If you have any desire (and you probably do) to use the API “in the future,” don't. Use it now.
Also, write tutorials! Just testing and building sample projects on the API, and then sharing what you do and learn, will offer the kind of exposure the project needs.
At some point sooner than later, I think it's incredibly important that the API goes in core, whether it's already broadly used or not. I think it's important for the API, of course, but also for WordPress itself. The API, I think, will define the next big chapter of WordPress's growth. But there are some things the core team and the community can do to make that happen.