The exact behavior of shields varies through the canon. It's generally accepted that shields block high-energy transmissions (phasers, disruptors, transporter beams), and "high-velocity" physical objects (photon torpedoes, asteroids, even smaller ships). However, there are also instances of a ship's shields holding back powerful but slow-moving physical barriers, such as the gate into the Dyson's Sphere in the TNG episode "Relics". So, one would think with shields at full that a collision with a piece from another ship would leave the debris worse off than the shielded ship.
In addition to deflector shields, most Federation ships have a deflector array. This array is highly configurable, and so can do basically anything the scriptwriters need, but its primary purpose is to push away large physical objects while the ship is underway; it's the ship's "icebreaker", shoving aside particulate matter, asteroids, etc while the ship is moving, especially at warp. It's supposed to work in much the same way as the full shields, but is generally lower-power in "shield" form, requiring a "beam" to push higher-mass objects. It's logical, given the difficulty involved in locking transporters on to a moving target, that the deflector array would simply be overwhelmed. The very large ship remnant that Enterprise specifically has to dodge is also half the size of the Enterprise herself; at some point the act of using any sort of beam to shove away a hunk half as heavy as yourself without any change in your direction or velocity would offend our Newtonian-oriented sensibilities.
The in-universe answer would probably be that the deflectors, both the full shields and the array, is simply overwhelmed by the sheer size and number of debris objects that must be shunted aside. Why they're still effective at near-full power after that, to absorb one of the Narada's missiles, is anyone's guess. The out-of-universe answer is that it's more visually engaging to see a piece of flotsam the size of an aircraft carrier gouge out a few panels of the brand-new Enterprise's saucer, with or without shields. Typical Rule of Cool.