Well, there are many issues that can cause a working site to not work when moved to a new server.
Is both servers Apache servers or Windows? Makes minor differences in set ups.
Is the PHP level the same on both servers? (PHP 5.3 as in your or PHP 5.5 on the new one, as example?)
Did you install Wordpress BEFORE moving your site to the new server? As Astonecipher mentioned, might
just be that you never installed Wordpress on the new server first.
Lastly, different servers hosts their default web server folders differently. Your error message showed this
folder name: ‘/home/z________/public_html’ On you new server is that the same? /home/z______/ is
a bit odd, but, the layout of each server must be exactly the same for it to work by just moving the site to
the new server. Otherwise, you need to alter references in your code to the base page folders…
Hope one of these helps!