Im am no expert in this topic, But I do clean metals often. I can only say what I would try. Most of the time the answer is just to polish and clean manually.
From what I can see, you have relatively hard water, and you tried to clean mineral deposits off the faucet. The Vinegar was the correct application, however you are left with some residual mineral deposit which was not removed by the acid wash (vinegar)
I would try now to clean the excess deposits manually. what you don't want to do is damage the chrome surface so steer clear from any sand papers or rough pads you should only use sponge or cloth. Also recommended is to use some toothpaste as your abrasive additive, it will have just enough grit to get the chrome clean.
Try to find a long thin piece of cloth, put a dab of toothpaste and wrap the cloth around the tap, then pull both ends one at a time to make a diy polisher. If you could get one person to "shank" the cloth (pull-pull motion) while you use the back of a spoon to press the cloth harder against the tap (at a point) you will get some really good polish action going.