Thanks for asking.

In general: there's no difference between the demo version and the 
full version except for the 30 day timeout. You're effectively trying 
the full version.

Regarding the issue you are having: 
If you have set in Remote Buddy > Prefs > General, that Remote Buddy 
should start automatically at login, it really should. Depending on 
what this setting is set to, Remote Buddy will take care of it being 
on or off the list of applications starting at login ("Login items"). 
You don't have to take any further action like adding RB to the Login 
Items list yourself.

If this option doesn't work for you as intended, please do the 
following:

1) Close System Preferences (if open) 
2) Launch Remote Buddy 
3) Ensure that the option is checked in Remote Buddy > Prefs > General 
4) Quit Remote Buddy 
5) Reboot

Apple did never provide a real API for adding and removing Startup 
Items except for two AppleScript snippets for this purpose. The last 
time I was dealing with this AppleScript code (which was in 2006 if I 
recall correctly), it performed unreliable for both adding and 
removing entries from the Startup Items list. For that reason, I 
decided back then that Remote Buddy should check, whether it is 
currently on the list or not and compare it with its setting. And if 
there's a difference, either add or remove itself from the list 
accordingly.

One more thing to pay attention to: please ensure that you have copied 
Remote Buddy to your Applications folder and that you start Remote 
Buddy from there rather than from a downloaded and mounted disk image 
(dmg) file.

If the problem persists, please let me know which version of OS X you 
are using and I'll be happy to work with you until the source of the 
problem is located and the problem is solved.

Best regards, 
Felix Schwarz