Temporal mechanics is an enormously complex subject due to the infinite inter-relationships between each object in a temporal continuum. Devices and beings that can manipulate time and the fabric of reality may also further interact with objects (or subjects, in the case of the players of Jumanji). Even so, there is a significant difference in how time plays out after the game is completed for Alex, in 1996, than it did for Alan and Sarah, in 1969, as well as in the very fabric of reality that the each set of players (Alan, Sarah, Judy and Peter from 1969 to 1995, and Alex, Spencer, Martha, Bethany and Fridge from 1996 to 2016) interacted with each other.
Assuming that winning the game, no matter how long after it started, resets the timeline for all players, we know (from Jumanji: Welcome to the Jungle) that Alan was sucked into a jungle within the game, while Sarah continued to live in the real world (cinematically). Peter and Judy joined the game and, by virtue of rolling a "5", drew Alan out of the jungle, a fabric of reality provided by the game. This is also where all the animals, the monsoon and Van Pelt originated, being pulled into the real world from the game's reality. Once the game was won, and time was reset, Judy and Peter would never be able to remember what happen, because the temporal mechanics removed their participation from Alan's and Sarah's real timeline; they all played the game in Alan's and Sarah's reality but in a timeline that no longer existed. Therefore, while Alan and Sarah, as originators of the alternate timeline, could remember, Judy and Peter would not remember, as time was reset before they would have played the game (and, in fact, before they were born). This is what also allowed Alan and Sarah to purposefully interrupt Judy's and Peter's timeline and sparing their parents with no consequences to themselves.
In contrast, Alex, in 1996, and the rest of the players, in 2016, were pulled into the game's reality from their own; everything that occurred was not in a separate timeline, but in a separate reality provided by the game. Upon winning the game and shouting its name, they were each pulled out of the game's reality and deposited back to their own timeline, memories intact. Alex, in 1996, allowed his timeline to play out, choosing not to prevent his future compatriots from playing the game (and who knows if that may have had disastrous results; he may have returned to the game to be stuck forever, or he might have created the dreaded time paradox).