Neural Networks can have a lot of free parameters (GPT-3 has 175 billion). Essentially, the limit is from the hardware on which they run. You can model very complex data - like text, images, times series data. And solve different tasks - multilabel classification, regression, anomaly detection, object detection, text and image generation.
Feature engineering was an integral part of doing (classical) Machine Learning to get good results. Neural Networks build features hierarchically using the raw data. They learn features from the data. Doing some feature engineering will most probably help - but large amounts of data and large models might do even better.
Training models on large text and image datasets adjust the parameter values using the data. In a sense, you store knowledge into the model. You can reuse that knowledge and transfer it to your problem. Use your custom dataset and do a little training (compared to the initial) using the model.
Doing Deep Learning boils do doing a lot of matrix multiplications. The best part is - you can do them in parallel. Modern GPUs allow for very effective parallel computation on a single, multi-GPU, or multi-node GPU setup.
Of course, nothing is perfect and Neural Networks are not an exception to the rule. They require lots of data to work, are notoriously hard to train and fine-tune correctly, and explaining predictions is a work in progress.