[Galaxy Note 4 Q&A, Help & Troubleshooting] andalism:
Some reports did compare..... and note 4 outperforms note 3 buy a huge margin.

1. Not all cameras have ois, not even the note 3. Note 4 has it and that allows for relatively long shutter speeds like 1/8s (I've seen note 4 do 1/4 once but that's rare). That's a huge upgrade from note 3.

2 . The note 4 uses sony's IMX 240 sensor which is 1/2.6 inch size. The note 3? And older generation sensor is of a smaller 1/3.06 size. Now. Note 3 is 13MP and note 4 is 16MP. So individual pixel size of both cameras is of the same 1.12 um size. However it's a known quantity that IMX 240 's has better iso performance than I'm 135. Thus note 4 should have better low light shots.

3.its been known that the note 3 takes bad low light shots due lack of ois and the old IMX135 sensor.

4. Almost all galaxy series engage in some kind of "image stacking" software to reduce noise in low light shots. Aka night mode. This is probably low light shots EXIF data is eroded .

As for the lag. What lag ? Jerkiness in viewfinder? Lag in auto focus?

From phonearena

" Outdoors at night, the Galaxy Note 4 soaks in quite a bit more light than the Galaxy Note 3, which results in more visible illuminated details at the expense of overexposure. Photos taken by the Galaxy Note 3 look decidedly darker and noisier in comparison. They are also much softer, while the Galaxy Note 4 retains fairly sharp detail. Low-light photography has seen a big improvement in the new phablet, and by dialling down the exposure setting a notch, you will emerge with more usable night photos than before. "

From Engadget

"But low-light performance is what impressed me the most about the Note 4. While ISOCELL helped improve the GS5 over the Note 3, it still wasn't very good. Optical image stabilization has made a tremendous difference here; objects that barely show up at all on other Samsung cameras can be easily seen on the Note 4. In fairness, the images still don't look as natural here as they do on the iPhone 6 Plus, but this is by far the best nighttime imaging performance I've seen on a Samsung phone."

If your note 3 does better low light shots than note 4. Your phone is probably faulty or you picked the wrong settings for the camera.

It's funny how so many people here expect unrealistic results from this camera. Is this camera better than a PnS camera? Nope.

This this camera worse? Not really.

If you're talking about junk market for PNS cameras. They use a 1/2.3 inch sensor which is not all much bigger than note 4.

A lot of them don't have OIS or Sony's newer image sensors ( Sony's sensors are widely used in many brands).Note 4's image quality is probably better

If you're talking about the mid segment. They use approx a 1/1.7 inch sensor which can probably do a little better low light shots. But they are not really cheap anymore. Eg canon s120. Olympus xz2 etc...

Note 4 loses out a bit here. But that's cause these cameras are meant to be an upgrade from smartphone cameras

If you're talking about about the premium segment. Eg canon gx7 gx1 Sony rx100. Most of them have 1 inch sensors which boast relative great low light performance.

(They still suck compared to mirrorless or dslrs though). However they are probably as expensive as your note 4 or even more.