Basically you have to decide what is most important to you. Do you want to have the widest range of content offerings(Roku), or the best compatibility between other Apple devices and software(Apple TV)?
The Roku box for example can play Hulu and Hulu Plus. The Hulu service is in very high demand and a very good content stream for the Roku. The Apple TV at this time will not play it out of the box, although it can be hacked to do so.
The biggest advantage that the Apple TV has is if you own or plan to use an iOS device in conjunction with it. For example, you can use AirPlay to stream music, photos, video, and apps to your Apple TV. This can be very handy in many situations. If you don't already have an iPhone, iPad, or the like - and don't plan on getting one anytime soon, then this shouldn't be a big draw.
The Apple TV also works flawlessly with iTunes. For example you can use iTunes Match then stream the audio to your Apple TV very easily. Roku can do some similar things with iTunes, but is does not have the same level of integration.
Overall, both devices have benefits and downsides, it just depends what you think your primary use will be and what hardware ecosystem you have. For more detailed information I would take a look at this review that compares the two.
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
No, not every iPhone app will be on the new Apple TV. Developers have to release apps specifically for the Apple TV, not just porting the iPhone app.
Yes, you can mirror the screen of any iPhone 4s or later to one of the newer Apple TVs.