What do we call something that assigns a weight to each case study which shows the importance of that case?

You are seeking a novel usage, so answers will necessarily be to some extent a matter of opinion. However, they will probably be drawn from a limited range of words. I restrict my suggestions to three. Other folk may add to the list:

assessor = a person whose job is to officially say how much something (such as a property) is worth especially so that it can be taxed according to that value law : a person who knows a lot about a particular subject and whose job is to give advice about that subject to a judge or other court official : a person whose job is to officially say how well someone has done on a test, in a competition, etc.

Merriam Webster

monitor = to watch and check a situation carefully for a period of time in order to discover something about it {in your case, its weighting}

Cambridge

evaluator = someone whose job is to judge the quality, importance, amount, or value of something {in your case, its weighting}

Cambridge

In giving a name to the neural network code you might therefore choose to anthropomorphize it and describe it as the evaluator, the assessor or the monitor.

It seems that your Neural Network (NN) is doing a ranking of the importance of the cases. Here, a rank is similar (if not identical) to the OP's concept of weight.

I searched on scholar.google.com for neural network ranking model and found quite a few hits.

A re-ranking model for dependency parser with recursive convolutional neural network. In this work, we address the problem to model all the nodes (words or phrases) in a dependency tree with the dense representations. We propose a recursive convolutional neural network (RCNN) architecture...

Neural Ranking Models with Weak Supervision. The reason may be the complexity of the ranking problem, as it is not obvious how to learn from queries and documents when no supervised signal is available. Hence, in this paper, we propose to train a neural ranking model using weak supervision, where labels are obtained automatically without human annotators or any external resources (e.g., click data).

So if you called your NN a ranking model, you would be understood by your target audience.

Consider a scoring network, which assigns a score to each input

score (MWD)

5: to determine the merit of : GRADE