The new Sites has a fresh, modern look and allows for more precise placement of content. As of yet, it is still missing some important features, most notably the ability to copy a site from a template. I hope that this will be addressed soon.
You can continue to switch back and forth from old to new through 2017 (see below), however in 2018, Google plans to no longer support the old editor. Therefore, if you are creating a new site, it is best to use the new editor at this point.
There is not currently a tool to migrate content from old sites to new sites. This would be a manual copy and paste for now.
I will update if and when Google comes out with a migration tool.