My Note 3 didn't need root but there have always been very annoying things about Android (to me, coming from the land of Apple) and required root to fix, and the rest just being nitpicky things.
My Note 3 when it got KitKat got a visual downgrade to the Galaxy S3 UI, and looked dated and lost the 'light effect' lockscreen effect, and was doing the S3's ripple water effect. i had also updated my Note 10.1 tablet to 4.4.2 and it got the new TouchWiz seen in the S5, which started to grow on me. i wanted that on the Note 3 but instead got a visual downgrade.
HyperDrive 8 gives me the S5 UI, most of the apps, even the weather widget and the nicer looking toggles.
The annoyance has been one since the early days of Android, the always-cluttered status bar up top. i never could stand how it looked filled with Facebook, email, various other icons, especially given the launcher i use having badge number support, they just came off messy and redundant. i preferred iOS's status bar where it was clean, with only the battery, signal, and wifi status showing at most, and the notification details showing only when you swiped down, and badge numbers on icons. thanks to Xposed, which requires root, i got that cleaner status bar. sadly, that Xposed module didn't seem to work on the stock ROM with just root which inspired me to try the above ROM.
Towel Root doesn't trip KNOX, mine is still at 0x0, but it does add in a 'custom' padlock icon on the boot screen which would invalidate any warranty just as KNOX tripping would. i haven't found any way to remove that yet.