The plasma membrane of a nerve cell and the internal membrane in a chloroplast perform very different functions, yet both are essentially made of phospholipid bilayers. How does the composition of these membranes differ to allow them to have different functions?

It's true that the plasma membrane of nerve cell and internal membrane are made up of phospholipids but they vary in protein associated/embedded with these membranes. Each protein perform a unique function which provide the overall unique reponsibility to a membrane according to its cell .

Different cells in body have different specialized plasma membrane to perform their specific function i.e. sarcolemma is the cell membrane of myosite, oolemma is of egg and axolemma is the specialized plasma membrane of axon. Axolemma is involved in generation of action potential because of its lipid bilayer that work in association with spectrin and actin (cytoskeleton components) and a trans membrane protein axolemma.

The inner plasma membrane of chloroplast need to perform separation of component and movement of substances, so it will have those particular proteins in its membrane. And in neuron, plasma membrane will have specific protein like for forming ion channels that are needed to transmit the signal.