To a layman like me, they both sound like similar concepts (though, of course, they're not!): The aggregate number of yards a QB had run with the ball in his hands before either passing off the ball or getting intercepted, etc. Is my understanding even remotely correct? What is the difference between the two?
Rushing yards are an official stat. When a player receives a hand off or a backwards pass, he then advances the ball by running forward or "rushing". Most often you see this stat in the NFL for running backs. However quarterbacks and receivers may also have rushing yards in a game. You may see updates during a game with a rushing yard total for players, it is the total rushing yards to that point in the game. Rushing the Football Wiki
Yard run usually refers to what happens on a certain play. I.e. "The player X had a 12 yard run". This means he would get 12 rushing yards also.
Announcers may refer to a yard run after a receiver catches the ball also. These yards after the catch are added onto his receiving yard total and the quarterback's passing yard total. Yards after catch (YAC)
Both Rushing yards and "yard run" will be measured from the line of scrimmage with a loss of yards being a negative yard run and subtracting from the rushing yard total.
If the quarterback passes the ball, he will not get credit for any yards that he has run prior to the pass (Pass meaning legal forward pass).
A rushing yard is a "running yard" gained on an official "rushing" play. This occurs when the quarterback either runs the ball himself (without passing), or hands the ball to a halfback or fullback for "rushing."
There are running yards that are NOT counted as rushing yards on other, non rushing plays. For instance, suppose a quarterback completes a pass for 20 yards, and the receiver runs another 15 yards. Those are 15 "running yards" that are not rushing yards; it counts as a 35-yard pass. Suppose a defender intercepts a pass and runs it back 10 yards. Those are 10 running yards that are not rushing yards. And a 25 yard kickoff or punt return are running yards but not rushing yards.