Panasonic ZS1 has a 10.0MP 1/2.5-inch (5.744 x 4.308 mm ) sized CCD sensor . On the other hand, Fujifilm X100 has a 12.0MP APS-C (23.6 x 15.8 mm ) sized CMOS sensor and features EXR processor.
One other difference between these two cameras that is worth mentioning is that Fujifilm X100's sensor doesn't have an anti-alias (Low-Pass) filter. Removing anti-alias filter increases the sharpness and level of detail but at the same time, it increases the chance of moire occurring in certain scenes.
Below you can see the ZS1 and X100 sensor size comparison.
Sensor Size and Resolution Comparison image of Panasonic ZS1 and Fujifilm X100 Cameras
As seen above, Fujifilm X100 has a 15.1x Larger sensor area than Panasonic ZS1. Larger sensors give photographer more control on the depth of field and blurry background compared to smaller sensor when shot in same focal length and aperture.