What is the difference between Cased-based Reasoning and Rule-based reasoning?
As said in the comments, there may be a part of subjectivity in the answers to this question.

Yet, I think it is fair to say that case-based reasoning mostly belongs to what is often called transductive methods, while rule-based (as well as neural networks and most statistical models) models mostly belong more to what is often called inductive approaches. 

In short:


Transductive methods will not try to induce a generic model from your past observations/experiences, but will simply try to predict/find the answer of a new problem by looking at past experiences. This means you have to maintain a memory of past experience, and then select the most adequate one to answer your new problem. The most well-known representative of such approaches is probably K-NN.
Inductive approaches will try to induce a "compact" model from your past observations/experiences, and to generalise (make an induction) from those. Once this is done, you can forget your past experiences and only use your learned model to make prediction.


The parallel you can make is between looking up a table of stored input/output values (e.g., to get a p-value), which is transductive, and using a mathematical formula to compute a new prediction, which is inductive. 

Lastly, it should be noted that the way you build your model when performing induction largely depends on your applications, available knowledge and needs (there are certainly ways to learn rule-based systems from data).