Is it important for customer service staff to know the difference between customer service & customer care ?

yes, because customer service is a core/crucial and very important arm of a firm which encloses customer care as one of its subsections.If a customer service staff is aware of this,then it helps in giving him/her a well detailed information of the job/profession and it also informs him/her that the first objective of customer service is customer care,after which others like sales,marketing and advertisement follows.thank you

Customer care is a unique product quality selling that,s you are providing to customer and customer services is the satisfaction of the customer on their visit to your branch.

Yes exactly, Because customer service is the service given to the customer to make them feel to come and visit your store again and customer care is the service given to the customer for their health and other issues. 

Yes because on customer service u have to give your best for your customer as your first priority moment they was in your hand and all thing that u need to do for them to feel tour customer care