Difference between 'lie' and 'lie down'

Solution 1:

The verb to lie describes the state of being in a horizontal position.

I often lie in bed for hours before falling asleep.

Why are you lying on the floor?

The verb to lie down describes the action of going from an upright position to a horizontal position.

You should lie down if you're feeling dizzy.

She lay down on the grass and went to sleep.

For the verb form differences between to lie (be horizontal), to lie (tell an untruth) and to lay (to put flat), see this article from Professor Pullum on Language Log.

Solution 2:

lie down is a phrasal verb, an expression and an idiom.
phrasal verb/ expression:

1 to place oneself or be in a prostrate position in order to rest or sleep

idiom:

2 to accept without protest or opposition (esp in the phrases lie down under, take something lying down)

lie ², on the other hand, is just an ordinary verb like any other.

to be situated, esp on a horizontal surface ⇒ "the pencil is lying on the desk", "India lies to the south of Russia"