Canon IXUS 165 has a 20.0MP 1/2.3' (6.17 x 4.55 mm ) sized CCD sensor and features DIGIC 4+ processor. On the other hand, Fujifilm XQ1 has a 12.0MP 2/3-inch (8.8 x 6.6 mm ) sized CMOS X-TRANS II sensor .
Canon IXUS 165's sensor provides 8MP more than Fujifilm XQ1'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 IXUS 165 and XQ1 sensor size comparison.
As seen above, Fujifilm XQ1 has a 2.1x Larger sensor area than Canon IXUS 165. 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.