Having seen the dog coming towards her, she quickly crossed the road. Seeing the dog coming towards her, she quickly crossed the road.
What are the differences between those two sentences?

Question

Having seen the dog coming towards her, she quickly crossed the road. Seeing the dog coming towards her, she quickly crossed the road. 
What are the differences between those two sentences?

Answer

The first sentence implies that she crossed the road after she saw the dog coming towards her.
 The second sentence implies she crossed the road at the same time as she saw the dog coming towards her.

Both sentences imply that her reason for crossing the road was seeing the dog come towards her, though :)

I hope that helps!