I am going to develope the 2D game in Iphone using Quartz.
what is the main Difference between Quartz and QuartzCore?
I have searched a lot over the internet
You should read the Quartz 2D Programming Guide. After reading the guide, take a look at QuartzDemo.
There are a number of very nice Quartz2D and CoreAnimation animation examples in this project, from the 360iDev conference session on Core Animation:
http://github.com/neror/CA360
I think you might be confusing a few things here. Quartz lets you do static 2-D vector drawing, but this is different from the QuartzCore framework, which you import into a project if you wish to use Core Animation. Core Animation is what you're looking for when it comes to 2-D animation on the Mac or iPhone (if you aren't going to go the way of OpenGL ES). You really do not want to use Quartz to redraw a moving scene, because you will get terrible display performance.
This does get confusing, because you can use Quartz to draw 2-D vector or raster art into UIViews or CALayers, and then animate those around using Core Animation. For a good introduction to Quartz, I recommend the Quartz 2D Programming Guide. For Core Animation, there is the Core Animation Programming Guide, as well as the Core Animation book by Bill Dudney.
You're right, there is a lot of sample code out there for Quartz on the Mac, as well as for Core Animation, but the nice thing is that Quartz and Core Animation use almost identical APIs on the Mac and iPhone. There are a few small differences, but for the most part code written for one runs on the other. As an example, we designed the Core Plot framework around Core Animation, using Quartz drawing, because we can have an almost identical codebase between our Mac and iPhone versions of the framework. That's at least one large piece of sample code that you can look at.
Finally, Opacity is a really neat utility that lets you draw vector artwork and generate the Quartz drawing code to reproduce that artwork within your application. It's a great way to try things out.