Typically people are using test-kitchen to test their recipes with integration tests against a local or remote vm/cloud. It uses either
chef_zero which right now are effectively the same thing. But to directly answer the question
chef-solo == chef-client -z == chef-client --local-mode
It really doesn't matter at this point which one you use,
-z is just the short option form for
--local-mode so those are truly identical.