You know in the sparse matrix, every item has four links (left, right, up and down)? In the version in the book, each item only has two links, up and down. The left/right links become implicit, because he stores the items contiguously in an array (so right of item 14 is item 15), and inserts "spacers" so you know when a row ends and the next one begins. This doesn't algorithmically improve the runtime, but it uses half the memory and gives it much better cache behaviour (btw: this is sometimes a charge levied at Knuth, he only cares about algorithms and not practical runtime when it comes to things like cache coherency. This is ludicrous, as this example shows.)

That's the change to the "basic" algorithm, but he then goes on to add a bunch of new features to basic Dancing Links, modifying it so that it's suitable for different kinds of constraint solving (adding required rows, things like that).

If you've implemented Dancing Links in the past, you owe it to yourself to check out the book version! He also has hundreds of really great exercises about it.