NN Making Poor Averaging Fit whilst LGBM Regressor Fits Perfectly
It looks like your neural network has a high bias. There are a couple of things you can do as a first step to attack this:

increase number of hidden layers
increase number of neurons
increase number of training epochs

Generally, neural networks have a lot of hyperparameters and so would be good to optimise using a gridsearch for example.
In terms of your first two models, it is of note that you need to double check they are not overfitting the training set.