Do 3rd party camera apps really improve picture quality? I've got a Samsung Galaxy Grand Neo (Jelly Bean). It has a 5 mp camera but I'm not satisfied with the picture quality. Will it help if I use another app like Camera Zoom FX or anything? If yes, then which app should I use?
Some camera apps improve the photos taken, but it's usually on a case-to-case basis. Some phones have limited / low-quality hardware and won't benefit from 3rd-party apps. Others have high-quality hardware, but poor camera software.

It's difficult to recommend a specific app. I have a Samsung Galaxy S4 and have installed the following: Camera Zoom FX, ProCapture, Snap Camera, official Google Camera. However, I don't get very good photos, probably because I think the camera is defective. My phone has been dropped several times already. 

But your mileage might vary. Try the different apps, and see what works for you. There are trial versions.