The main difference between an Edge-lit LED and LED-backlit display is exactly as it sound:
Edge Lit: Edge lit TV has all the lights turned on the edge of the screen to light on TV. With the introduction of LEDs in this model allows a much thinner profile than normal for a TV back lit.
Back Lit: LED-back lit television has to inform all the display with a rear view of the TV. With the introduction of LEDs in this model, a higher contrast ratio and deep black, with a technology called local dimming.