Eigenfaces VS Deep neural network for classification
As it stands, neural networks outperform the Eigenfaces approach. With this I mean that neural networks can solve a large range of problems with more accuracy. 

Eigenfaces are nice because they can work already with a small amount of training samples, specially compared to neural networks that are known to be data intensive. So, in a small amount of data setting, you could start with Eigenfaces for feature extraction and pair it up with an SVM for classification. However, now a days, there are pre-trained neural network models (e.g. Facenet), which offer pre-trained embeddings for you to use for feature extraction. Then, all you need to do is train "a last layer" for classification. Bottom line is, you don't need to start from scratch, making it obsolete to use Eigenfaces for feature extraction.