Battery
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Nexus 5 has a
2300 mAh battery and though there have been many complaints regarding low capacity, a single day of moderate usage is totally possible. Not to mention there is already an update being rolled out to fix all that so this concern about battery life should soon die away
Android 4.4.3 spotted in the wild, coming soon to fix battery draining camera bug|androidandme.com
Xperia Z1 on the other hand has a beefy
3000 mAh battery so there should be no doubt about the battery life
Performance
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Nexus 5 is heavily supported over at XDA and basically it is a
Developer Phone and not really for the masses. There are literally tons of modifications available for your phone and the life of a Nexus device is always more than that of a non-nexus device. You can tweak almost every aspect of your phone to suit your requirements. And performance, really, is just one aspect of it.
Xperia Z1 isn't left out when it comes to unofficial support. There are some good ROMs and Kernels for this device though nowhere near that available for a Nexus. The customization is the only area where this device will lose out to a Nexus but its stock ROM is pretty good to boot.
Now as you can see that both these devices are literally same spec to spec in terms of CPU,GPU and RAM, there is nothing to differentiate if they operate right out-of-the-box. However, throw in a bit of tweaks like overclocking, undervolting, memory optimization....and you get the idea. The raw performance should theoretically be the same.
Looks(aesthetics)
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This is a purely subjective issue so I will not go further but if you ask me, I can't really decide. Though I like the minimalist approach of Nexus 5 but the sharp corners of Xperia Z1 are no less spectacular
I think it is a tie
Verdict
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Nexus 5 if you want performance tweaks
Xperia Z1 for
microSD card slot,
water resistance,
shock resistance and
FM Radio
Sony Xperia Z1 vs Nexus 5 comparison review: two high-end Android smartphones slug it out|PCAdvisor