Samsung introduced new improved keyboard in Android 5 lollipop but has not made it available for the Galaxy Note 3 Neo. Fortunately XDA has made it available by porting it to Note 3 Neo.
To use it, you must download and flash the Note 4 Mini V6 ROM on your Note 3 Neo and then flash it. Download Samsung Android 5 keyboard from here and flash it using Note 3 Neo custom recovery but if you had earlier installed the Note 4 Mini v6 ROM, you must be having the custom recovery already installed. The ROM is very efficient and will improve your battery life as well.
You will find multitude of enhancements on this Samsung keyboard compared to the one you already have on your Note 3 Neo. It feels much polished and better to type on.