All the above terms can get confusing. Let's start from the top.
Approach Shoe is a general term used by retailers to identify a type of shoe. In reality an approach shoe can be just about any type of shoe, but will include the best type of shoes for rock scrambling.
So, Approach Shoe is the heading you would click on when shopping online.
Sticky Rubber is just a marketing term that more and more retailers are using. It can mean anything.
Stealth Rubber is an actual type of rubber found on many types of Vasque shoes. Many hikers swear by Stealth Rubber, including me. It will improve your rock scrambling by 25%. There are different types of stealth rubber. The best for sandstone is the kind with the red oval on the sole. See the above photo. More information on Stealth Rubber
Gryptonite is another type of rubber found on shoes. I have worn hiking shoes with Gryptonite and found them to be as sticky as Stealth Rubber. Here's a link with more information: http://www.montrail.com/technologies_gryptonite.aspx As you will read, there are different kinds of Gryptonite.
FriXion is a third type of sticky rubber out on the market. I have not worn shoes with FriXion, but suspect they are on par with Stealth Rubber. For more information.
If you do a lot of rock scrambling, I highly recommend buying shoes with Stealth Rubber or Gryptonite.