[Varyon]

Independent essentially functions as though you were continuing to use Concentrate its entire duration, without requiring you to do so, but you have to define what exactly it's doing during that time beforehand. For TK, I'd say you could tell it to keep striking some foe with a TK'd sword, or attempt to grapple and pin a particular foe, and it will do that - but nothing else.

Persistent just holds things at a steady-state of sorts for its duration (10 seconds by default). However, while Independent only allows for a single instance to be active at a time, there's no such limit for Persistent - provided you have the duration, you could literally have hundreds of instances running at once with no issues. Once you've lifted something into the air, you can just leave it there and move on to the next thing, relying on Persistent to keep it up.

With Independent, you can use TK to lift a shield and order it to stay in front of you and Block attacks that come at you, without using a Maneuver each round to keep it up, but you can't use TK while that's going on. With Persistent, you could lift a piece of cover in front of you and have it stay there, then go on and use TK on other stuff, but that cover wouldn't move with you or actively intercept attacks.