I'm not sure what you mean by applying a NN to linear regression.

I'll try wording it differently. Sometimes a NN can outperform linear regression on regression problems, like in the example if there's a nonlinear relationship between some features and car price. But neural networks are also prone to over-fitting so I recommend against having a NN as one's first attempt to model some data. I recommend starting simple and trying complex models when it gets difficult to improve results in simple models.

I didn't say this before but another benefit of starting simple is that linear regression is usually much faster than neural networks, so you can iterate faster and try out more ideas quickly.