Old rule: Coin tokens can only be removed from a player's Coffers in that player's Buy Phase (or when
instructed by a card such as Butcher); they cannot be used when buying a card via the promotional card Black Market.
New rule: You can spend Coffers at any time during your turn.
Some cards had to change to make this change possible:
Old Merchant Guild: While this is in play, when you buy a card, +1 Coffers.
New Merchant Guild: At the end of your Buy phase this turn, +1 Coffers per card you gained in it.
Old Patron: When something causes you to reveal this (using the word “reveal”), +1 Coffers.
New Patron: When something causes you to reveal this (using the word "reveal") in an Action phase, +1 Coffers.
The main reason for this change is that it makes play easier. You no longer have to plan ahead in your buy phase, calculating what exactly you want to buy and spend the Coffers for it first. The interaction with Black Market always surprised people, and it's now gone.
Impact: some cards get stronger because of this change. In my opinion, the most notable winner is Storyteller, now allowing you to spend coffers for draw. It's also nice for Spices and Silk Merchant, allowing you to use the newly gained Coffers immediately if you have multiple buys. Black Market is a bit stronger now, and of course Merchant Guild changed the most.
First of all, Merchant Guild used to say "While this is in play", and it no longer does. This is a lot stronger when you play it multiple times with e.g. Throne Room, because now you'll actually get the effect multiple times. It's weaker when you put it into play without following its instructions, for example with Way of the Pig or an opponents Enchantress attack.
Second, it used to say "when you buy" and now it says "per card you gained". This can make a difference if you gain a lot more cards than you buy in your Buy phase. Likely candidates are Border Village, Livery, Stonemason, Port, Experiment, Death Cart, Talisman, Masterpiece, Fortune, Rocks, Duchess and a lot of Events.
Third, you used to get the coffers immediately, and now you'll get them at the end of your buy phase. You couldn't spend the coffers immediately anyway so usually this changes nothing. But be careful of the combination with cards that get you out of your buy phase without ending it (Villa, Cavalry).
Implementation: You can now spend Coffers in the client by clicking on the Coffers symbol, like Villagers. This works whenever nothing else is happening (so we're not halfway resolving some other card). Also very much like Villagers, you will almost never have to do this. At convenient times to spend coffers the old slider to spend them is still there, and the 'autobuy' feature will automatically spend them for you.
A number of cards have 'stops' build into them where the client pauses and asks you explicitly if you want to spend Coffers at this time: Storyteller (right before drawing), Wine Merchant (right before possibly discarding from the Tavern mat), Fortune (right before doubling your Coins), Basilica (right before taking 2VP), Swashbuckler (right before you may get the Treasure Chest), Black Market (right before buying)
There used to be three things in Dominion that could happen between turns. Donate, Mountain pass, and choosing what type of extra turn to do next if you had multiple of those coming. Now the first two are gone, and only that last remains. This is nice, because now nothing can happen anymore outside of a turn, which used to lead to some weird edge cases that now no longer exist.
Old Donate: After this turn, put all cards from your deck and discard pile into your hand, trash any number, shuffle your hand into your deck, then draw 5 cards.
New Donate: At end of turn, put all cards from your deck and discard pile into your hand, trash any number, shuffle your hand into your deck, then draw 5 cards.
Old Mountain Pass: When you are the first player to gain a Province, after that turn, each player bids once, up to 40D, ending with you. High bidder gets +8VP and takes the D they bid.
New Mountain Pass: When you are the first player to gain a Province, at end of turn, each player bids once, up to 40D, ending with you. High bidder gets +8VP and takes the D they bid.
One important thing to notice is that both cards used to happen after we check if the game ends, and now happen before we do so. This implies that if the game ends, they now will happen, while in the old situation they didn't.
On boards where you'd love to lose cards at the end, like with Wolf Den, Wall, Bandit Fort or Curses, ending with a Donate can now save you where it used to be useless. Also be extra careful of the interaction with Hunting Grounds or Catacombs because the end-game-check comes right after.
If you manage to empty 3 piles in a game where no one has scored yet, ending the game with a single Province on a Mountain Pass kingdom can be very disappointing: your opponent gets to bid 40D first. Better buy a Duchy or Estate.
Playing & Replaying Cards
new rule: An effect that tries to play a card for the first time can only do so when the card is where the effect expects it to be. If it can play it the first time, subsequent plays will also work.
old situation: There was no generic rule restricting trying to play a card.
This change doesn't affect all that much, it's mostly a new rule to allow card wordings to be less clunky. Vassal playing Village Green will no longer allow you to double play it.
Old Prince: You may set this aside. If you do, set aside an Action card from your hand costing up to $4. At the start of each of your turns, play that Action, setting it aside again when you discard it from play. (Stop playing it if you fail to set it aside on a turn you play it.)
New Prince: You may set aside (on this) a non-Duration Action card from your hand costing up to $4. At the start of each of your turns, play that card, leaving it set aside.
Prince now also is a Duration. The change implies you can now use Throne Room to play a Prince and not be disappointed. It also means the Princed card doesn't hop into and out of play anymore, and the loop can no longer be broken by failing to set something aside. Also Prince itself now will be in play, which can be nice with Pilgrimage.