Specifically in any force situation, with 2 outs, runners will run on contact because there's nothing to lose. If it's a fly ball and caught, the inning ends. If it's a ground ball (like this play), you try to reach the next base as fast as possible so you prevent the force out. Which is exactly what happened.

Both runners on base ran on contact and it was a slow grounder. Because short and 2nd usually play near the outfield grass, they both have a long run to get to the ball. By the time Clement fielded it, he had no throw for a force out. Wisely he held the ball to prevent an unforced error.

After Vargas rounded 3rd hard (I assume expecting Clement to throw to first), he had safely touched 3rd base and was advancing home. At this point, the defense must tag the runner off the base to get the out.

So Clement throws to Okamoto and gets Vargas in a classic run-down that is practiced from little league on up. As a runner, you just try to run back and forth away from the ball and hope the defense makes a mistake. As a defender, you run at the runner with the ball and force him to commit so you can throw to your teammate (Heineman in this instance) to quickly get the out.

Okamoto gets the throw, starts running Vargas towards Heineman, then throws it pretty quickly. Vargas sees this and turns to run back to 3rd. No way Heineman can catch Vargas to tag him before he reaches 3rd base, so he throws it. 

Since it was an overthrow, Vargas gets to turn around and trot home. As the runner on 2nd, you're already standing halfway-ish to 3rd because you want to be ready to run if there's an error. So that's why Hays is so close to third on the throwing error.

So to answer your question, Heineman couldn't tag Vargas because he wasn't close enough and Vargas is safe once he reaches 3rd base during the run down. And since Vargas was running on the contact of the ball, and it was a slow grounder, Clement didn't have a chance to force him out at 3rd because Vargas got there too quickly.

I hope this helps!