In "Best of Both Worlds" part 1 act III, the Borg hail the Enterprise and demand the personal surrender of Picard:
"Jean-Luc Picard, captain of the starship Enterprise, registry NCC-1701-D, you will lower your shields and prepare to transport yourself aboard our vessel."
Picard refuses, and the Enterprise fires on the Borg vessel. The Borg respond by capturing the Enterprise in a tractor beam, which also completely drains the Enterprise shields. Then -- instead of beaming drones over to capture Picard -- they start attacking with a cutting beam. Commander Shelby instructs Data to rotate the phaser frequencies, which disables the Borg cube and allows the Enterprise to escape to a nearby nebula.
This is a completely different tactic than what the Borg do in act IV. The Borg fire charges into the nebula, which forces the Enterprise out, where it is again captured by the Borg tractor beam. This time -- instead of using the cutting beam -- the Borg board the Enterprise, kidnap Picard, and head toward Earth.
The Borg use the cutting beam again in part 2 act V. However, this occurs after Picard has been kidnapped back by the Enterprise crew. Picard is still part of the Borg consciousness at this point, so their objective to make Picard one with the collective has already been fulfilled. They have different goals in this act.
Why do the Borg bother with the cutting beam in act III, instead of simply kidnapping Picard like they do in act IV?
The Borg's goal is to capture Picard. They outright state it in the quote above, and actually follow through with it in act IV.
They're not interested at this time in destroying the Enterprise, disabling its warp drive, or assimilating its crew. They had an easy chance to do these in act IV, but instead they headed to Earth.
Enterprise shields do not seem to be an impediment to the Borg's transporters. Their tractor beam drains the ship's shields, and this question shows that the Borg can transport through Starfleet shields anyway.
The out-of-universe answer is that it adds time to the episode, adds dramatic tension, and gives Shelby an opportunity to order the phaser frequency rotation. However, I am looking for an in-universe answer.