What are the practical implications of ciphertext distinguishability? Commonly there are four ways to "break" a secrecy-focused cryptosystem:


Recover the secret key
Recover the message
Distinguish an encryption from random noise
Distinguish the encryption of two different messages


The practical implications of the first two breaks are obious: They allow arbitrary message recovery (more or less) fast.

However I'm unclear about the practical implications of the the last two breaks. 

So let's assume an encryption algorithm or a public key cryptosystem which fails to provide indistinguishability but also doesn't allow key or message recovery. As an example assume the scheme from the "Present an attack for the combination of OTP and textbook RSA" question under a chosen plaintext attack.

So:
What are the practical implications if a distinguishable cryptosystem would be used in practice?
This isn't really a "hard" answer, but an attempt to give some intuition or motivation.

One can interpret indistinguishability as an overapproximation of the most common notions of security: Any system that is broken in a more practical way will also fail to meet indistinguishability, that is, all practically important security requirements are in fact subsumed by indistinguishability. It is therefore a sensible and defensive approach to require indistinguishability. This proceeding also agrees with the expectations one usually poses to a "good" system: It should be as close as realistically possible to Shannon's definition of perfect secrecy.

On the other hand, a system may well be distinguishable but do not admit any relevant attacks — however, in practice, distinguishing attacks are often the first sign that a system may be about to break in more severe ways. A potential reason for this could be the fact that the structure of a system needs to be well-understood by researchers to find such attacks, and then other weaknesses can become visible as well. I suppose the most prominent example is the RC4 cipher, which continued to accumulate more and more known biases (breaking indistinguishability) over time until it had to be considered insecure for practical purposes.