There are artificial and biological neural networks. The artificial one is based upon the biological one. Therefore dropping the 'artificial' or 'biological' seems fine to me.
This is practical, however I strongly disagree with this semantically. The term should always be accompanied with 'artificial' or another name should be invented as they are vastly different fields. Search Neural Network, I assure you that you will no longer find biological information; it is littering.
At this point the name is well established. If it makes you feel better, just remember to tag "artificial" on to the beginning, nobody will mind.