As of 2023 (question is now 4 years old), NFL and NCAA rules have been modified and are closer than ever regarding the clock stop after a play. Why was the NFL rule modified? To speed up the game and minimize the waste of time during loooong games (source):
According to the rules, if the player is making forward progress and he goes out of bounds with no member of the defense edging him, then the play is ruled out of bounds. If the defense attacks him and he, on purpose, goes out of bounds, then his forward progress is marked on the field while the clock keeps running.
Every minute in the NFL is crucial, and the rules were made so that no time is wasted, especially during the last minutes of the game. Most of the time, the clock stops once the ball is out of bounds and the game resumes as soon as the referee places the ball on the ground.
You can watch an official video provided by the NFL about "rule 4 - section 6".
The NCAA article "Fewer clock stoppages on first downs and more 2023 college football rule changes" explains this new rule in order to maximize play time:
Why is this important?
Originally, the game clock was stopped when a team gained a first down and the clock would resume when a referee placed a marker on the field and the team was awarded the first down. With this new rule change, for coaches, it is important to maximize each play, without the usage of possibly calling a time-out, and think of ways to score on each drive.
Excerpt from "Why Does The Clock Run After Out Of Bounds In College Football" of Game Day:
Typically, during both college football games and NFL games, the clock will stop until the next snap in the game whenever there is an incomplete pass, the team is timed out, or the play ends with a score. But, there are instances in which the clock is stopped while or after a player is out of bounds.
The moments in which the clock will stop while a player is running out of bounds only occur during college games whenever there are under 2 minutes left in the first or second half of the game. The clock will stay stopped for that out-of-bounds runner until the next snap is issued.
This is also explained in other news reports, such as Sporting News:
Running clock after first downs except for the last two minutes of each half
This is like the NFL with the exception of the final two minutes. It also is the most drastic change to the game. Previously, officials would stop the clock after each first down. ("College football's new clock rules for 2023, explained: 3 changes aim to shorten games")
NCAA 2023 Rules Book -- NFL Rules Book