fall / fall down

In my opinion,  if something falls it is moving downward. But if something falls down it is usually a) Not intended to have fallen and b) Will/ or Has come to an inevitable stop. I.E. Landing on something 'The rock fell.' the rock moved downward 'The rock fell down' the rock moved downward and landed on something
                  
The bird had fallen from a nest and The bird had fallen down from a nest.

The bird cannot fall  from a nest, so using fell down is a stylistic choice for a writer/speaker. There is no real difference between fallen from a nest and fallen down from a nest.

To fall means: to move from a higher to a lower level, typically rapidly and without control:

Bombs could be seen falling from the planes

To fall down means:  to be inadequate or unsuccessful; fail:

The deal fell down because there were a number of unanswered questions.

Rahul should have made clear that his definition of "fall down" represents an idiomatic usage and is not the only acceptable meaning.

As Tolley points out, "to fall down" is redundant. Nonetheless, it is often used for emphasis. Being redundant is a stylistic issue, not a grammar error.