They compete in many areas such as tablets where the iPad competes with several Google Nexus and Android-powered devices. Then in the mobile space, there had been rumors that Apple had considered switching its default search engine to Microsoft's (MSFT) Bing. Of course, the #1 area where they compete is mobile where Apple and Google Android-powered phones have been able to become dominant players. Apple has been going after a few of the smartphone makers legally such as Motorola, Samsung, etc. and while they have not won everything, they did land a few big wins. They have yet to go after Google directly though.
They have also been outbidding themselves in recent patent sales in order to launch more lawsuits, etc.
Then recently, with the launch of the iPhone5 came the maps fiasco where Apple was basically unable to secure renewal rights and had to replace Google maps with its own offering in what is now clearly a big mistake as the product was clearly not ready. It's hard to blame Google on this but it was well played. Why? Because in the end, Google launching its own iOs map was a huge success with over 10 million downloads in the first few days (including yours truly).
Are They Playing This The Right Way?
I'm not convinced that Google and Apple are playing this the right way. They currently dominate an exploding market, have been able to basically eliminate Research in Motion (RIMM), fend off attacks from Microsoft, Amazon (AMZN) and others, etc. But they keep waging war against each other, spending billions in lawsuits, and trying every possible tactic to destroy each other.
There's A Better Way
Why wouldn't they choose to collaborate to some degree, to make sure that they keep control of this market? It seems like that would make a lot more sense, especially in a fast growing market. They would achieve much more by keeping similar market shares in a fast growing market than by risking the arrival of a major competitor. Don't you think?
When Kodak announced it was going to sell a significant patent portfolio, it was hoping to have giants such as Microsoft, Apple and Google engage into a bidding war that would have generated a ton of money. Instead, Apple and Google agreed to join forces and present a joint bid. Incredibly smart move and it looks like they will save several hundred millions because of it.
I personally think they need each other because no government would let only one of them take control of the mobile space. So while they can't "officially be working together to eliminate competition", I certainly think they should collaborate a lot more than what they are currently doing.
Disclosure: Long Apple