[Galaxy Note 3 Q&A, Help & Troubleshooting] DrawnToLife:
Are you aware of how camera sensors work?

Smart stabilization is meant to offset the use of higher ISOs/lower shutter speeds so that your pictures turn out less noisy. Given that the Note 3's max aperture is f/2.2, let's make an example.

In the daylight, your ISO can drop and thus picture quality can improve (this is very barebones, but it's for conceptual purpose). In essence, higher ISO = more noise, especially the smaller the sensor size. You can visibly see that the Note 3 has a very small sensor. In fact, even on APS-C cameras, noise performances starts to suck around ISO 3200~6400. ISO is meant to make each pixel brighter (or something to that effect), and the smaller those pixels are, the more noise you'll generate (again, not exactly, but that's the gist of what you're experiencing). So, with 13 megapixels fit onto the small sensor, you're likely going to start seeing noise at like, ISO800 (this is a random guess, but it's probably true).

Anyways, so during the daytime, there is a lot of available light, so the ISO can be reduced and exposure can be adjusted using shutter speed (assuming aperture stays the same). This improves picture quality as lower ISOs generally equate to less noise. However, as you get later into the night, less available light means that one of two things has to happen. Either your shutter speed gets slower to let in more light, or your ISO cranks up to become more sensitive to the available light (and thus more noise). Usually a combination of the two occur to get a trade off between quality and shutter speed.

A quick browse on google gave me, 1/15, f2.2, ISO 1000, as EXIF data from an iPhone 5s taking a picture at night. As you can see, the ISO is pretty high for the small sensor, and the shutter speed is quite low. As a rule of thumb, you generally want at least an equivalent shutter speed to the focal length, but given the crop factor of this lens, I have no idea what that'd actually be. But 1/15th is very, very hard to hold without some form of IS/OS, even on full-frame cameras. What you're experiencing is this effect. The low shutter speed to let in more light means that even if you breath and shift the camera 1cm, you'll get blur. It's not out of focus, but the subjects weren't in the same place because you moved, causing them to be rendered in shift. The noise is the result of the ISO being too high; the pixel sensitivity isn't that great, and so you're getting all kinds of weird colours that the phone is trying to represent without definitive data. Again, we're assuming that the aperture stays wide open under these conditions to let in the most light.

I hope you got the answer you were looking for. Basically, what you want is pretty hard to do, even with a full-framed DSLR (although it's becoming less true with recent image processing). You can't really turn of IS/OS and expect the pictures to be great. There's a lot of other things that are taken into account behind the scenes that are usually beyond your control on your phone. Smart stabilization using image processing algorithms to help mitigate the impact of higher ISOs and lower shutter speeds by post-processing the images you take on the fly. Such is the cause of the delay.