How to open Emacs like in Linux Terminal
If the emacs you are running in Linux creates a new window then you are running it as an X11 application (a GUI application).  Normally emacs on Linux and other systems just runs in the terminal you have open (as a console application).

The version of emacs that ships with OS X is compiled just to run in a terminal. To replicate running in a new window you need another version of emacs. I would use Aquamacs which is an emacs modified to act like a normal OSX Cocoa app. The standard GNU emacs has a Cocoa app version which defaults to be more similar to behaviour on other platforms and is available in a binary version here.

You can also get X11 versions as well.  Here is an exhaustive list of the available types of emacs for OS X.