Strut tower bars work to keep equal contact patches on both the left and right tire when cornering. During straight line driving they don't really do much. So in a sense it would be wrong to say it makes the chassis stiffer. Lemme try to explain a lil, and then you can agree or disagree.
If you create a parallel line from the one wheel hub to the other and then go up the shock dampers and then create another parallel line from the top of one damper to the other you get a trapezoid.
For a car without a strut tower bar, turning during cornering distorts the trapezoid which results in one of the wheels (inner wheel usually I believe) having more contact patch than the other.
When a strut tower bar is installed, it keeps the geometry of the suspension constant (or that is what it is supposed to do anyhow, some cheapo STBs may not work so well) and keeps both tires equally contacting the road.
The result with the STB is proper geometry of the suspension during turning and that's why the cornering will feel more crisp, as well as reduce bodyroll in the front.
I guess I could draw a picture if needed but I think I explained it pretty well.
