Zou and Hastie in their paper proposing the method give the following explanation:
In this paper we propose a new regularization technique which we call the elastic net. Similar to the lasso, the elastic net simultaneously does automatic variable selection and continuous shrinkage, and it can select groups of correlated variables. It is like a stretchable fishing net that retains ‘all the big fish’. Simulation studies and real data examples show that the elastic net often outperforms the lasso in terms of prediction accuracy.