Difference between sklearn.neural_network and simple Deep Learning by Keras with Sequential and Dese Nodes?
I don't think they will give you the exact same results as the internal implementations for 2 same operations are different even across pytorch and tensorflow.
What makes Keras a better option is the ecosystem. You have the DataLoaders which can load the complex data in batches for you in the desired format, then you have the Tensorboard where you can see the model training, then you have preprocessing functions especially for data augmentations. In TF/Keras, you now even have data augmentation layers, in PyTorch Torchvision provides this in Transforms. Then you have the flexibility, you can define what types of layers in what order you want, what should be the initializer of the layer, do you want batch norm between layers or not, do you want a dropout layer between the layers or not, what should be the activation of hidden layers you can have relu in 1 layer and tanh in other, you can define how your forward pass should exist, blah blah. Then you have the callbacks to customize the training experience, and so on.
Given ,  sklearn.neural_network  and simple Deep Learning by Keras with Sequential and Dese Nodes, are the mathematically same just two API's with computation optimization?
Yes Keras has a Tensor Support and could also liverage GPU and Complex models like CNN and RNN are permissible.
However, are they mathematically same and we will yield same results given same hyper parameter , random state, input data etc ?
Else apart from computational efficiency what maker Keras a better choice ?