Canon 600D has a 18.0MP APS-C (22.3 x 14.9 mm ) sized CMOS sensor and features Digic 4 processor. 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 6MP more than Canon 600D'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 600D and D5300 sensor size comparison.
As seen above, Nikon D5300 has a 1.1x Larger sensor area than Canon 600D. 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.
DxOMark Sensor Scores
DxOMark is a benchmark that scientifically assesses image quality of camera sensors. It scores camera sensors for color depth (DXO Portrait), dynamic range (DXO Landscape) and low-light sensitivity (DXO Sports), and also gives them an overall score. Canon 600D and Nikon D5300 sensors have been tested by DxO and the results show that D5300 has a better overall score of 83, 18 points higher compared to 600D's score of 65.