The difference originates from Apple's legacy in desktop publishing and graphic design. The nice thing about the Apple algorithm is that you can lay out a page of text for print, and on screen, you get a nice approximation of the finished product. This is especially significant when you consider how dark a block of text looks. Microsoft's mechanism of hammering fonts into pixels means that they don't really mind using thinner lines to eliminate blurry edges, even though this makes the entire paragraph lighter than it would be in print.
The advantage of Microsoft's method is that it works better for on-screen reading. Microsoft pragmatically decided that the design of the typeface is not so holy, and that sharp on-screen text that's comfortable to read is more important than the typeface designer's idea of how light or dark an entire block of text should feel. Indeed Microsoft actually designed font faces for on-screen reading, like Georgia and Verdana, around the pixel boundaries; these are beautiful on screen but don't have much character in print.