It's not that clear to me the difference between the 2 rigs, and the action, tried both and noticed that on both cases the worm stays up underwater

 

What exactly is the advantage to use one or another rig? they look and feel the same to me

Depending on the shaky head...some make the worm stand up better, others are football or “rugby” shaped to make it come through rock better. Bullet weights do not excel under those conditions. Bullet weights are better for weeds. 

Shaky head is best on a horizontal plane; TR is best more vertically penetrating cover.

I prefer a shakeyhead over a t-rig most of the the time as can feel the bottom better with the line tied to a jighead then with a threaded-through weight.  I will use a t-rig when:

 

1. I want to fish a larger plastic with a small amount of weight and desire a little separation between weight and plastic, but the free rig is taking more and more of this job
2. When I want to quickly fish soft cover and want max snag resistance.  I will often build a "t-rigged jig" for this and peg the weight
3. When I want to use a bead and crack the plastic along the bottom.