In your first code snippet, you don't assign any value to $addme before you test it. If one of your post variables is called addme, then the value of that variable will be assigned to $addme if the PHP configuration option REGISTER_GLOBALS is turned on. This option should never be turned on, as it opens a major security hole. I'll leave it as an exercise for you to research this.
The $_POST array is a much, much safer way to access POST variables. The above research should explain why this is the case.
So, finally to answer your intial question, PHP can operate differently on different servers because it gives the server admin the right to configure things in a way that suits them, regardless of how inadavisable that configuration is.