As i known, the full edition is server+viewer function, host edition only server function? But how about the capability of server function?
TeamViewer Host gets installed on e.g. a Server as a service, and you can connect to it from remote anytime you need. Normal teamviewer full edition connects people by "running" the teamviewer software themselves (having to run them manually each time), or to connect to TeamViewer Host servers.
Have a look here, which describes the editions: http://www.teamviewer.com/en/download/index.aspx