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While it's an Internet Wikipedia, it does a good job of explaining what a sway bar does as well as some of the drawbacks (way better than I would put into words).
As for the heavy springs, no bar vs. light springs, heavy bar, that all has to do with spring rate... replacing one method (springs) with another (bars) with the Bars being the easiest / quickest to swap and adjust.
Take a look at the 'downside' portion of the wiki in particular.
One would think so, but IMO there are clear differences.
Assuming identical wheel rates between the stiffer springs/no bar vs softer springs/big bar, you still have a few differences:
Softer springs/big bar may = same wheel rate when the car is rolling side to side, but the soft springs will exhibit more forward weight transfer under braking. And conversely, more rearward under acceleration. Too much movement here can cause quick unloading of the front/rear of the car (esp under braking) and can be skittish at the limit. A good driver can compensate for this, however.
Big bars tie the left and right sides of the suspension together. As one side rolls, the other side also compresses. You lose some of the independent suspension qualities here--not ideal for bumpier circuits. However, when the car turns into a corner (under trail braking), the front will roll...and the bar will tie the unloaded side to the side which is rolling...which actually lowers the front of the car during this phase. This behavior can increase front traction in this portion of a corner.
The above is assuming all things being equal. But--for example, one can combat the forward/rearward weight transfer of softer springs by introducing more anti dive and anti squat into the chassis to compensate.