There are several ways and steps to achieve this.
First you need to create 2 grid module, or 1 extra module if you are using quickstart package.
As you have stated both modules are showing same content, lets understand about how grid positions work.
Grid positions are simply same as module positions and it will load modules in whatever position you enter as grid position. So basically grid position must match your module position.
So if modules are in following position,
Module Position: header
Then Grid position will be : header
If module position: bottom1 and you have 4 modules in this module position
then grid positions will be:bottom1 or bottom1[1] or bottom1[2] or bottom1[3] or bottom1[4]
Now we are clear on grid position setup, you need to setup your second grids position names to match your modules so we get different content in second grid module.
if you will be using same grid layout but different content then simply copy current grid module.
Make changes to grid positions as explained above.
Now assuming both modules are setup and enabled.
If you have a menu item then simply use menu assignment options in module settings to show grid module on appropriate page.
If you don't have a menu item then create an empty k2 item and create a new menu item linking to this k2 item. Then assign your grid module to show on this new menu item.
We might need to hide empty white space with a css code.
Although it seems like a long procedure it is quite quick solution.
Let me know if you need further help on this.