I cast Azusa, Lost but Seeking in my first main phase. I play two additional lands. Second main phase, I sacrifice Azusa to Ashnod's Altar and use the mana to pay for a second Azusa. I CANNOT play two additional lands, because the game knows how many lands I've played this turn, even though the second Azusa was not around to see them.
I cast Rule of Law. I CANNOT cast additional spells this turn, since the game knows how many spells I've played this turn, even though Rule of Law was not around to see them.
I cast Muldrotha, the Gravetide in my first main phase. I cast a creature card from my graveyard. Second main phase, I sacrifice Muldrotha to Ashnod's Altar and use the mana to cast a second Muldrotha. As I understand it, I CAN now cast a second creature card from my graveyard, even though as we've seen the game knows how many spells I've cast from my graveyard this turn, even if this Muldrotha was not around to see it.
This seems backwards to me. Why does Muldrotha work differently from Azusa and Rule of Law?
Your land drops for a turn is something baked into the rules of the game. Azusa simply increases this number. You had 1 available land drop before Azusa, 3 after she entered the field, 1 again when she left, and 3 when a new one entered. She is not giving you permission to play a land; she simply alters the rule (while she is around) to increase the number).
Rule of Law is similar in that it simply creates a limit that didn't otherwise exist.
In both cases it is adding to, or subtracting from, something that is already allowed by the rules (playing lands/casting spells).
Muldrotha is different again in that Muldrotha offers a permission that didn't previously exist. That permission, much like Karador, Ghost Chieftain, is tied to the permanent because it isn't changing an existing rule. It is allowing you to do something you couldn't otherwise do. So, you cast a card with Muldrotha with the permission it gives you. The permission for that card type is used up because it is something Muldrotha tracks. Muldrotha leaves. A new Muldrotha enters and gives you a new permission to cast that card type since it is a new Muldrotha and this Muldrotha's permission hasn't been used yet.
Or, in other words, the game can track things you do normally (casting spells/playing lands) so anything that cares about how many of each occurred this turn can see it. Muldrotha doesn't care about that. Muldrotha simply says "you are allowed to do this thing" and Muldrotha will track it since it isn't something the game cares about since you can't normally do what Muldrotha is allowing. So, when a new Muldrotha comes along, a new permission exists.
I am sure someone else can come along to explain it better, but hopefully this provides some insight into the differences.
Effects that add to the number of lands a player can play in a turn (such as found in Azusa, Fastbond, and Exploration) are governed by C.R. 305.2a-b. These rules are what makes this kind of effect special, and don't apply to other effects that allow a player to play cardsplay lands or cast spells of a certain kind in a manner not otherwise allowed (such as found in Muldrotha or Crucible of Worlds).
Rule of Law's ability could alternatively be read as "Each player who has cast a spell this turn can't cast additional spells" (compare with Ethersworn Canonist).
EDIT (May 5, 2020): Edited in view of update with Ikoria: Lair of Behemoths.
I cast Rule of Law. I CANNOT cast additional spells this turn, since the game knows how many spells I've played this turn, even though Rule of Law was not around to see them.
I cast Muldrotha, the Gravetide in my first main phase. I cast a creature card from my graveyard. Second main phase, I sacrifice Muldrotha to Ashnod's Altar and use the mana to cast a second Muldrotha. As I understand it, I CAN now cast a second creature card from my graveyard, even though as we've seen the game knows how many spells I've cast from my graveyard this turn, even if this Muldrotha was not around to see it.
This seems backwards to me. Why does Muldrotha work differently from Azusa and Rule of Law?
Rule of Law is similar in that it simply creates a limit that didn't otherwise exist.
In both cases it is adding to, or subtracting from, something that is already allowed by the rules (playing lands/casting spells).
Muldrotha is different again in that Muldrotha offers a permission that didn't previously exist. That permission, much like Karador, Ghost Chieftain, is tied to the permanent because it isn't changing an existing rule. It is allowing you to do something you couldn't otherwise do. So, you cast a card with Muldrotha with the permission it gives you. The permission for that card type is used up because it is something Muldrotha tracks. Muldrotha leaves. A new Muldrotha enters and gives you a new permission to cast that card type since it is a new Muldrotha and this Muldrotha's permission hasn't been used yet.
Or, in other words, the game can track things you do normally (casting spells/playing lands) so anything that cares about how many of each occurred this turn can see it. Muldrotha doesn't care about that. Muldrotha simply says "you are allowed to do this thing" and Muldrotha will track it since it isn't something the game cares about since you can't normally do what Muldrotha is allowing. So, when a new Muldrotha comes along, a new permission exists.
I am sure someone else can come along to explain it better, but hopefully this provides some insight into the differences.
play cardsplay lands or cast spells of a certain kind in a manner not otherwise allowed (such as found in Muldrotha or Crucible of Worlds).Rule of Law's ability could alternatively be read as "Each player who has cast a spell this turn can't cast additional spells" (compare with Ethersworn Canonist).
EDIT (May 5, 2020): Edited in view of update with Ikoria: Lair of Behemoths.