What are the pros and cons of a Turbo Trainer over Training Rollers?
Stand trainers

Pros


no concerns of falling over/slipping off, especially for certain types of training, e.g. one-legged drills
better for interval/spike training, where transitions in speed are not smooth
more options for variable/controllable resistance


Cons


less natural riding style because of bike's fixed position
may develop pedal stroke imbalances (or allow them to go undetected)
requires special mounting skewer (most brands)
can cause significant tire wear


Rollers

Pros


more natural style; bike responds more like actual riding
can help develop smoother pedal stroke and better form


Cons


easy to fall off and cause damage (to body, bike, and surrounding objects)
fewer options for variable/controllable resistance (only a few makes/models offer this and they're expensive compared to options for stand trainers)


Many folks I know have both a stand trainer and rollers and focus their training using the relative strengths of each type, i.e. stand trainer for "power" workouts, and rollers for "form" training.

As to your specific concern about natural movement: there is at least one stand-type trainer that is somewhat more natural than a fixed stand. It is considerably more expensive and occupies a lot more space than a normal stand trainer, though.