I also had problem with free version 1.2.12 when moving my site to new host. Same setup software and hardware and so on. Only newer version of joomla, I updated to latest version 1.5.23. I could not get ver. 1.2.12 to work, it would not show agent image in "bottom" module, just and image-X for image missing, the title was there when I put it to SHOW TITLE - YES. Also in the component in stats it would only show bots, nothing else and thoose bots would be both boots and normal webusers (I recognized some ip's).
Never the less, I uninstalled vers. 1.2.12 and installed older free version 1.2.9 and it came alive right away, just like on the old server. So for now I'll use this version... because I really like joomlawatch