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The battery is probably because one has a better screen or a faster processor as for your issues try performing a system repair
Xperia C has a very inferior MediaTek processor and a low-res display vis-a-vis XSP's Qualcomm S4 Pro and HD Reality display.Obviously battery life will be affected.
But I get around 2-3 days on a single charge.I use DeepSleep Battery Saver Pro.
Just try it,it may work for you.