When should I use Backdrop CMS rather than Drupal 8?
Backdrop CMS is a good choice if:


You are currently spending a lot of time managing your configuration using Features module. Backdrop includes built-in Configuration Management tools that allow you to deploy configuration across environments using JSON files, which should save a lot of time compared to Features.
Your needs are met by the current selection of Backdrop Contributed modules or you are capable of porting modules to Backdrop (it takes a few hours per module generally speaking). References from: Lullabot, Tag1, and Backdrop documentation.
You are familiar with Drupal 7 code and would like to transfer that knowledge to a newer system that will be supported in the future. Backdrop code is nearly identical in concepts, though there are several new additions to manage configuration. Drupal 8's code base is significantly different in many fundamental concepts.
You have an existing Drupal 7 site that you would like to upgrade. Backdrop includes an upgrade path (via update.php) that can upgrade your existing Drupal 7 site to Backdrop. Drupal 8 does not include an upgrade path from Drupal 7; the plan is to provide migration tools to move from Drupal 7, but at this point, it is not yet working. Backdrop does not yet have a port of the Migrate module, but it probably will in the future (since porting modules from Drupal 7 is fairly trivial).
You want to upgrade/build a site right away. Backdrop 1.0 had been out for about 2 months now and is stable. Drupal 8 may be coming out some time later this year (though we can only estimate the date). Drupal 8 does not yet support upgrades even between beta versions, so if you build a site on it now, you won't be able to upgrade it when the final version comes out.


Other features that Backdrop includes:


Backdrop is slightly faster than Drupal 7 out of box because of it's lighter core. Generally it's about the same speed as D7 but since D8 is slower than D7, Backdrop is faster than D8.
Backdrop has a built-in Panels-like module (Layout) that replaces the old Block system. Blocks are given new powers like contextual information, advanced conditional selection, multiple instances, control over markup, and config-based settings.
Updated libraries (e.g. new jQuery versions).
Responsive, HTML5 elements, and better mobile support out of the box.
All the other features listed on the Backdrop CMS homepage.