Sony HX350 has a 20.0MP 1/2.3-inch (6.17 x 4.55 mm ) sized BSI-CMOS sensor and features BIONZ X processor. On the other hand, Sony HX1 has a 9.0MP 1/2.4-inch (6.104 x 4.578 mm ) sized CMOS sensor and features Bionz processor.
Sony HX350's sensor provides 11MP more than Sony HX1'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.
Below you can see the HX350 and HX1 sensor size comparison.
Sony HX350 and Sony HX1 have almost the same sensor size so none of them has any significant advantage over other in providing control over depth of field when used with same focal length and aperture.