On a serious Note-
Second description has a tone that is geared at speaking to players reading the book and how the DM is likely to respond. It also goes into more detail of what is going on in Edit- Exploration mode. This is a player facing description (and more interesting by far)
First description has a tone that is matter of fact and is mostly neutral in player/DM facing.
Sort of. The second text starts with player-facing description (“Between encounters, your characters explore the world...”) then shifts to DM-facing description, starting from “Follow these steps to run the game in exploration mode,” and then returns to player-facing from “The Dungeon Master decides whether or not something you try actually works” on. This is a bit strange to me because it’s from the 4e DMG, so I would expect the whole thing to be DM-facing, since it’s not a book players are really expected to read.
In contrast, the first text is largely neutral, describing play from the perspective of an impartial observer, rather than directly addressing the players or the DM. This is also an interesting choice since it is from the 5e basic rules, and nearly-identical text is also found in the PHB. Both documents that are primarily for the players, but are expected to be red by both players and DMs.
One major difference that stands out to me:
4e said:
Some tasks involve a skill check or an ability check, such as a Thievery check to pick the lock on a chest, a Strength check to force open a door, or a Perception check to find hidden clues. Characters can perform other tasks without any check at all: move a lever, take up a position near the entrance to watch for danger, or walk down the left fork of a passage.
This suggests that it is the nature of the action that determines whether or not a check is required - picking the lock on a chest is an action that inherently requires a Thievery check, while moving a lever inherently does not.
In contrast, the other passage says:
5e said:
Sometimes, resolving a task is easy. If an adventurer wants to walk across a room and open a door, the DM might just say that the door opens and describe what lies beyond. But the door might be locked, the floor might hide a deadly trap, or some other circumstance might make it challenging for an adventurer to complete a task. In those cases, the DM decides what happens, often relying on the roll of a die to determine the results of an action.
If circumstances make the action challenging, the DM determines what happens,
often relying on the roll of a die to determine the results. This suggests to me that picking a lock might or might not require a check and it is the DM that determines whether or not a check is required, not the nature of the action. Further reinforcing this contrast is this bit from the 4e passage:
4e said:
Decide what you want your character to do and tell the Dungeon Master. The DM tells you to make a check and figures out your chance of success (a target number for the check).
So, in my assessment, 4e and 5e follow the same basic play loop of:
1. DM describes environment
2. Players declare actions
3. DM determines results
4. Repeat
But, there is a subtle distinction in how step 3 gets resolved. In 4e, the DM determines what kind of check, if any, is needed, sets the difficulty, and interprets the results. In 5e, the DM determines the results themselves, calling for a check if necessary to help make that determination.