You would not need to give up your US citizenship that is correct, however your mother would have to have a current Spanish citizenship, not sure if she has to live in Spain at the moment. You would have to apply on basis of your mother's citizenship, however there might be a little problem because she is not longer living in Spain, and I would think would have to give up her Spanish citizenship to get US citizenship. However to end all your questions, contact the Spanish consulate, and ask them what they tell you. To be honest it is not very easy to move between US and Europe, even though they are both largely first world countries/states.
I believe it is a process that if approved will take 1 - 2 years, if mother only US citizen now, she would have to apply first to become Spanish again, and after that you would be able to apply, 2 to 4 years. But to be fair to most PPRUNE members, they would not know to much about this, Spanish consulate would be the best step first.
By the way if you get told you must give up one of your citizenships, think long and hard, Spanish wages are probably some of the worst in Europe.