Can a headphone amp help the sound reproduction of even "fairly cheap" (~$100) headphones?
What can a headphone amp help with? When do I need one?
The most common use for a headphone amp is to feed multiple sets of headphones from a single headphone socket. They can also provide some amplification if your original headphone out signal is too weak.
See this Sound on Sound article from Martin Walker for a helpful explanation of when you might need a headphone amplifier. Also, check the Wikipedia article on headphone amps to understand the two different ways the term is used.