Oh you are so VERY wrong. Steve never wanted to be anything like Bill and Apple is very different from Microsoft. (Whether Apple is better or not depends on how you measure better.)
Apple has no interest in dominating any market. Apple just wants the high-end of every market because that's where the high-margin business is.
Compare this to Nokia. Nokia invested heavily in call-phone making capacity so they could mass-manufacture the lowest-cost cell-phones enabling them to capture the most market-share... they now sell more cell-phones than anyone else, but these are mostly the low-cost low-margin cell phones sold to low-income people in 3rd-world countries. Apple couldn't care less about that portion of the cell-phone market. Apple's iPhone is clearly targeted at the very top-end of the cell-phone market where it can command a lot of media attention as well as high profit-margins. Sure Apple wants to dominate the smartphone segment of the cell-phone market, but this segment will always be a tiny fraction of the overall market.
Apple never wanted to dominate the entire PC market either. They've only wanted to build the best quality (and highest margin) PC's that would dominate the top-end of this market. When Job's wasn't at Apple, they tried to follow a different business strategy by allowing 3rd parties to make PC hardware of their own design that would run MacOS... this was a strategy designed to try and "beat" Microsoft at their own game. Jobs was smart to kill that strategy as soon as he returned to Apple. He knew that he could only build the best products if he controlled the entire design process (or, at least, the vast majority of it) and that if he had the best products he could command the highest prices/margins. Under Jobs, Apple will never compete on price; they always compete on design and product quality. That's true whether they're going after a high-priced segment or a low-priced segment and its VERY different from HP or Dell or myriad others.
Jobs' vision is harder to see and understand in the PC marketplace than it is in the cell-phone marketplace. This vision is all about excellence and superiority in product design and manufacturing quality... and using this superiority to command a slightly higher price and higher margin than competing products. He doesn't want to kill those competing products, he loves and embraces those competing products because they still have to turn a profit (albeit on slimmer margins)... so they create a price-support under his own high-margin products.
A monopolist wants to kill his competition. Jobs embraces his competition.