Difference Between Original Wii Remote and Wii Remote Plus (RVL-036)

"New Play Wii Remote" is not an official term. You likely mean Wii Remote Plus (RVL-036) vs the Original Wii Remote (RVL-003).

Feature	Original Wii Remote (RVL-003)	Wii Remote Plus (RVL-036)
Release	2006 (Launch)	2010 (Bundled with Wii Sports Resort / Red/Blue)
Motion Sensing	Accelerometer only (tilt, shake)	Built-in Wii MotionPlus (Gyroscope + Accelerometer) — 1:1 precise tracking
Speaker	No	Yes (Audio output for game sounds)
Size / Shape	Slightly smaller, rounded bottom	Slightly longer, flatter bottom, larger A button
Colors	White only (Launch)	Black, Blue, Red, Pink, White, Zelda/Golden, Mario/Red
Connector	Expansion port (Nunchuk, Classic, MotionPlus dongle)	Expansion port (Nunchuk, Classic) — No dongle needed
Battery Life	~30h (Alkaline)	~30h (Alkaline) — MotionPlus draws slightly more
Sync Button	Red (under battery cover)	Red (under battery cover)

Which Do You Need?

Wii MotionPlus Required Games: Wii Sports Resort, Wii Play Motion, The Legend of Zelda: Skyward Sword, Red Steel 2, Tiger Woods PGA Tour 11+, Grand Slam Tennis. Original Remote + Dongle works but is clunky.
All Other Games: Both work identically.
Wii U: Both supported. Wii Remote Plus recommended.
Dolphin Emulator (PC): Real Wii Remote Plus + Bluetooth (Toshiba Stack / Windows Stack) + Sensor Bar = Best accuracy. Original + Dongle = Laggy.

Look at bottom: "Wii MotionPlus INSIDE" printed near expansion port = RVL-036 (Plus). No text = RVL-003 (Original).

Nintendo Wii