I’m frankly not sure how many people, even inside baseball, use the term “run-and-hit” anymore… but the difference was explained to me at a young age. 

     On a “hit-and-run,” there is usually nobody out, or sometimes one out, and the runner at first base breaks for second base as the pitch is made. The batter is required to swing at the pitch, no matter where it is -- his goal is to hit the ball on the ground, preferably toward the position vacated by the middle infielder covering second (right-handed batters usually try to hit the ball toward where the second baseman plays, and left-handed hitters try to hit it to the shortstop position). The best result has the batter punching the ball on the ground through the vacated hole, and there are runners at first and third. An acceptable result has the batter grounding out, but the runner makes it safely to second base on the play.
     A “run-and-hit” occurs in a similar situation, but with three balls in the count. On this play, the runner breaks for second base, and the hitter is expected to swing, IF the pitch is a strike; if it’s not a strike, it’s ball four, and the runner is entitled to second base anyway. (With two outs and the count 3-and-2, it’s not really considered and “run-and-hit,” because the runner will be going from first base automatically.)
     Occasionally – no, rarely – you’ll see a team pull a hit-and-run or run-and-hit with runners on first AND second base. Not often the best idea, for a number of reasons: a big inning is brewing anyway, so why waste an out to move the runners up?; the catcher has two places he can throw to gun someone down if the pitch is missed; and, if there’s no one out, the offensive team is setting itself up for a line drive triple play.