The diagram in the article shows five areas where the two companies theoretically overlap. First, operating systems. The claim is that Google's new Chrome OS will compete with OSX, but I think the markets for the two are entirely different. OSX is a fully-functioning desktop operating system, able to run native applications, surf the web, anything, really. Chrome OS, on the other hand, is, as I understand it, supposed to be a lightweight, web-based operating system. It won't run applications that are necessary for a lot of professionals, particularly in the design business, and won't really replace a full operating system like OSX or Windows; rather, it will be lightweight and perfect for portable computing, namely, netbooks. I don't think anyone's going to be giving up the functionality of OSX for Chrome OS; rather, one might have both operating systems, Chrome on a netbook, and OSX on a main computer.
The second area of overlap is music. Google lets you search for and listen to music, and this article claims that competes with iTunes, but, really, the two aren't alike at all. Streaming a song or two is nice if someone gives you a recommendation, but it doesn't replace iTunes' functionality, creating your own playlists, and, most importantly, owning your music, and allowing you to load it onto an iPod and have it anywhere you go!
In the video arena, Youtube and iTunes don't overlap much, either. Youtube is nice if you want to quickly find a music video, or one of the countless viral videos of people doing stupid and possibly dangerous stunts and finding an audience online, but iTunes video is more about movies and TV shows, high-quality, portable ones, which isn't something Youtube comes close to replacing.
Browsers is an area where they might actually compete, with Chrome and Safari, but I'd argue that's not such a big deal; neither browser is the main business of either company.
Mobile platforms are probably the biggest competition between the companies. The iPhone is undeniably awesome and popular, and Google's Android platform smartphones are increasingly popular, too. However, this brings me to a point: Apple's main business is hardware. They sell computers and mp3 players. Google's main business (where 97% of its revenue comes from) is advertising. Much of what Google does is aimed towards earning more out of advertising. Hardware does not enter the picture. It's more about the software, the web applications (and downloadable applications), and the vast majority of those don't compete with anything Apple has to offer. I'd say the two companies can coexist safely, at least for now. Of course, if you disagree, I'd love to hear it!