The changes to the rules affect when the golf ball is at rest. This means the ball is stationary somewhere on the hole and is essentially the time between the previous shot you played and the next one.
When the ball is at rest and it moves, if it was more likely than not that you were the cause which moved it, that is 50.1% sure, then it was seen for the purpose of the rules as being your fault.
However, the rule has now changed and you are only considered to have caused your ball to move if it is almost certain (at least 95% likely) or known that you moved the ball.
Sometimes you are left searching for your ball, especially if it has landed in deep rough, bushes or trees. In previous rules there was a one stroke penalty if you moved the ball when looking for it, even if the movement of the ball was accidental.
This rule has changed and now, when searching for a lost ball, you do not have to worry if it moves during your search as you can just replace it without penalty should it move.
If your ball moves while off the green and it was not 100% clear where the ball was initially at rest, you can replace it using the closest estimated place of rest. For example, if your ball half-buried in the first cut of grass off the fairway, replace it so it is in the same half-buried position as close to the original spot as you can.
The previous rule stated you should drop the ball when you were not sure of the location and this could have resulted in a different lie condition to that of previous lie.
These are not rule changes which make a huge difference to most people during a round of golf but you never known when you might need them.