"Google is a company that does software, but it's not defined by it. So why does Google seem to be aping Apple's every move?"
I think this guy is smoking crack. Not defined by software? Since Google doesn't sell any hardware, the only things customers see from Google -- and have EVER seen from Google -- have been interfaces to Google software. Apple may be a company whose chief distinguishing characteristics involve software differentiation, but Google is a company that does nothing but offer software.
Incidentally, offering Chrome (the Google browser) as evidence of competition with Apple is disingenuous. Chrome's rendering engine is WebKit, which Apple deliberately open-sourced (if Apple wanted a closed-source alternative, it could have done one). There is a very strong reason Apple benefits from others using the WebKit rendering engine: the more people use it and other genuinely standards-compliant rendering engines, the more incentive web developers have to adhere to standards instead of coding to meet the expectations of nonstandard, closed-source, oddball renderers (like MSFT's) that tend to prevent free access to all comers on the web. Google's launch of a browser with both new innovations (separate processes for each tab for memory recapture and application stability) and battle-tested tech like WebKit help dislodge bad nonstandard competition as a plausible de facto standard. Apple benefits from Chrome.
To the extent Android poisons the well for MSFT in the mobile space, Apple also wins from Android. After all, Apple will not license MacOS to third party handset developers, so this battle for non-Apple handsets must be left to someone else. There will be non-Apple handsets, and Apple benefits if MSFT doesn't control them; MSFT control of an ecosystem has been proven to work against the entire free world, and is the enemy of profit outside MSFT. Apple benefits from Android.
Whether Schmidt should go depends what Apple intends to get from Schmidt's involvement with Apple. If Schmidt is (as the article indicates) recusing himself from discussions that involve conflicts, then Schmidt may still offer very valuable communication and insight (and cooperation). (There is an antitrust doctrine that a company cannot conspire with itself, and hearing from its Board cannot therefore be an antitrust violation by a company. However, interlocking Board membership can lead to antitrust concerns involving the two different companies. Some kinds of aid and information from an outsider might be considered evidence of a conspiracy to restrain trade, that is, a scheme to prevent MSFT from competing with Google and Apple, companies on whose Boards Eric Schmidt and Arthur Levinson both sit.)
Whether Schmidt's usefulness to Apple has come to a close is a strategic question that depends on the strategic purpose of his involvement on the Board, and not on whether both companies offer standards-compliant browsers, desktop APIs that don't involve Win32, or a mobile operating system that doesn't involve licensing fees to MSFT.