The camera for WP 8.1 I believe is the same as the camera for WP8 at this point.
Seeing as WP8.1 is only in SDK and not officially released.
Feature wise Nokia camera is the better choice.
The default camera is ok for quick uncomplicated photos.
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I prefer nokia camera in my Lumia 920 as i have options to change the white balance, ISO, the focus etc, whereas in default app i dont have. Although i have observed sometimes the built-in camera app has taken better pics (not very often).
I will go for nokia camera , since I can play with manual controls , shutter speed and focus , I can also record video at 720p at 24fps which gives a cinematic effect in videos.
I own a L720 as well and I mostly prefer Nokia camera over default camera app. Nokia camera lets you capture a pic with serious amount of details in them. And also you get to tinker with various settings like iso, exposure time and white balance. Though I hate the time taken for processing a pic. I occasionally switch to the default app as it does not have any delay between shot to shot time. The pics taken from the default app are pretty good actually(after the black update). The color reproduction is nice and also the pics have good amount of detail in them(not as good as pics taken using Nokia camera though).