Panasonic ZS1 has a 10.0MP 1/2.5-inch (5.744 x 4.308 mm ) sized CCD sensor . On the other hand, Nikon D5300 has a 24.0MP APS-C (23.5 x 15.6 mm ) sized CMOS sensor and features Expeed 4 processor.
Nikon D5300's sensor provides 14MP more than Panasonic ZS1's sensor, which gives a significant advantage in real life. You can print your images larger or crop more freely.
On the other hand, please keep in mind that Max sensor resolution is not the only determinant of resolving power. Factors such as the optical elements, low pass filter, pixel size and sensor technology also affects the final resolution of the captured image.
One other difference between these two cameras that is worth mentioning is that Nikon D5300'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 D5300 sensor size comparison.
As seen above, Nikon D5300 has a 14.8x 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.