Regular chess is played on an 8x8 grid of squares, and pieces occupy one square at a time. They move along grid lines and diagonals and there's no such thing as being "halfway" between two squares. Every piece on the board has an X-coordinate between A and H, and a Y-coordinate between 1 and 8.

In this version, the pieces are no longer constrained to having integer coordinates. A pawn can be "halfway" between, say A2 and A3, or a queen could be sitting roughly in-between several squares at once. In regular chess, pieces are captured when an enemy piece moves to the same square as them. In this version, since the grid doesn't exist any more, pieces are captured when an enemy piece physically overlaps them.

It's kinda like normal chess is a low-resolution game with only 64 possible locations, and this is a higher resolution version of it where you aren't limited to just those 64 locations and can slide continuously all across the board.

In chess, piece locations are defined in terms of squares on the 8x8 grid.

In this analog version, locations are defined continuously, the checkerboard they move over is meaningful only to establish starting positions, and then is purely aesthetic.