There are restrictions on Airplay in Mountain Lion. It has to do with your video card that comes on the machines that you are using.
Your video card has to be powerful enough to do all the heavy lifting of streaming so that your CPU is unhindered and therefor you can smoothly still use your machine.
This question and answer should help you out and explain things and find an alternative, such as AirParrot.
Here is an Apple Support Article on the subject, listing the restrictions and giving more information on how Airplay Mirroring works.
Here is the excert from the article showing the compatible computers:
iMac (Mid 2011 or newer)
Mac mini (Mid 2011 or newer)
MacBook Air (Mid 2011 or newer)
MacBook Pro (Early 2011 or newer)
As for a Mac Pro, they are not listed as a compatible model that supports Airplay Mirroring, as discussed in this article here.
You can still connect a cable and mirror in that sense, that has not changed in ML. The mirroring feature of using a separate display is a completely different process than Airplay Mirroring.
I hope that this answer helps you out.
Best Answer
You have an airplay icon, not an airplay mirroring icon on a 4th generation iPod. You need a 5th generation to get mirroring. On the iPad, there is an option for turning mirroring on and off. So on my iPod 4 I get sound but no video on my TV when using Netflix, for instance. On my ipad I get both sound and video.