Suppose Saturn simply vanished. How would that affect the rest of the Solar System, specifically Earth?
The rings and moon system would disperse, but everything would remain in orbit around the Sun; nothing in the Saturn system is moving fast enough to break orbit, at least not immediately. Some moons might be gravitationally captured by other moons. The long-term interaction of the former moons could be quite complex, and close interactions just might eject some of them into more eccentric orbits. Titan itself contains most of the mass of the system outside Saturn; I guess its orbit would remain fairly stable. It's quite possible that Titan would "clear its neighborhood" of the other moons, and thereby qualify as a planet in its own right (it's certainly big enough).
All massive bodies exert a gravitational pull on all other bodies, so Saturn vanishing would have some effect on every other body in the Solar System, but the effect would be minor. Jupiter and Uranus would be most strongly affected, but both would merely have their orbits shift very slightly from what they are now. It would require some careful observations from Earth-based telescopes to detect the difference.
Over long time frames (millions or billions of years), the gravitational influence of planets on each other is more significant. For example, simulations suggest that Jupiter and Saturn formed closer to the Sun and migrated outward, and that one or more gas giant planets might have been ejected from the Solar System early in its history. Removing Saturn would change the long-term dynamic behavior of the remaining planets in ways that you would need some high-powered mathematics and computer simulation to predict. As far as I know, the orbits of the planets are reasonably stable over the next few billions of years; losing Saturn shouldn't change that.