HTC one is quicker at taking photos. But its gotten slammed because it can only do 4MP. You cannot zoom in too much with a 4MP photo or lets say a 8MP allows you to zoom in twice as much as a 4MP. The G2 like the samsungs is good when light is adequate, but when light drops noise creeps in especially with video. Z1 is an all rounder, good in low light as well as regular. G2 might be sharper because denoising is more agressive like samsung but when the light drops that agressive denoising works against you.
I've compared photos of a 5S with the Z1. At 20MP there was more noise in the photo but at 8MP and low ISO i would say the photos were equivalent. The biggest difference is between superior auto and manual. In superior auto, sometimes the wrong scene mode is picked, say like soft snap that can make a picture look wierd. That is why manual with iso selection produces the most consistent shots. Roule of thumb is 50-100 outdoors, 200-400 indoors. Going up to 800 if the subject is moving so as to get a faster shutter speed. Superior auto i've found tends to pick a higher ISO than could be got than with manual. Which means a more noisy photo. Follow the tips mentioned in this guide.
You said you tried many settings but this is a crap shoot. you need to tailor the settings to the subject and shot at hand. What are you trying to photograph and what is the lighting like. You've not mentioned the subject so far.
If you want to pursue this further can you put up the originals of the S3 & the Z1 so we can take a look at them.
