The last major rule changes to chess involved Italian differences in castling and en passant. Under Italian rules, a player could place the king and rook on any square between the two when castling. As
George Gossip's "The Chess-player's Manual' (1862) described it:

"It must be borne in mind that the Italian method of Castling, in which the King and rook have the choice of any of the intervening squares, as well as those they originally occupied (subject to the restrictions of not giving check, nor allowing the King or Rook to take up a position attacking any hostile man when performing the operation of Castling) was calculated to impart far more animation to an otherwise tame opening than the comparatively dull mode of Castling now practised in the rest of Europe ... "

George Walker (1803-1879) had been a staunch advocate of adopting the Italian style of castling as related in his 1832 book "The Art of Chess-Play: A New Treatise on the Game of Chess."

"I was formerly of opinion, with Sarratt, that it would improve Chess to introduce the Italian method of castling into this country. I own I am now less sanguine as to the probability of this measure's enhancing the interest of the game. Certain Gambits would become forced won games ; and the consequence would be, the second player would evade, and not accept them. In fact, in most Italian circles they now castle as we do ; and as taught by Greco and Cozio. Were Greco's Calabrian innovation not considered an improvement, it would hardly have been so generally adopted."

In 1880, according to Italian chess historian Carlo Alberto Pagni, the Italians finally abandoned their "free castling" rules and adopted en passant, abandoning their rule of Passar Battaglia which stated that a pawn could use its initial first move to avoid being threatened by an opponent's pawn in an adjacent column (essentially negating en passant).

The most recent rule change that I can come up with is the modification of the rules regarding pawn promotion.

In 1862 the rules of the British Chess Association allowed for a promoted pawn to be "exchanged for a Queen or any other piece his player may choose", and allows for the pawn to remain a pawn.

In the International Chess Code (which declares itself a revision of the British Code) from 1918 this rule has been modified to require promotion, list the pieces that can be promoted to (to exclude the possibility of promoting to a king) and to include the phrase "a piece of its own color". My understanding is that this rule was changed in the 1883 version of the code, but I can't find a copy of that online to link to.

There are some interesting scenarios that have been concocted where this rule difference can be the difference between a successful checkmate or a draw.