Even when you don't generally care about the properties digital signatures provide you almost always need authentication w/ encryption because block cipher modes make the behaviour of the encryption differ in surprising and important ways from an idealized black box.
Concretely:
Imagine I setup a service that accepts commands through encrypted messages with a static key. Only authorized parties know the keys.
Each (encrypted) command sent to the services includes a user and password.
At first glance you might consider this setup reasonably secure but if authenticated encryption is not used a MITM can modify the ciphertext to cause predictable modifications to the plain text, preserving the user and password but changing the command or its parameters.
Eg. Imagine the cipher used is AES-CTR, the attacker can just xor whatever he wants into the plaintext. If a CBC mode is used, the attacker has a little less flexibility but still a lot.
Beyond breaking control the lack of authentication can also break confidentiality: For example, if the service provides different error behaviours for different corrupted inputs, I could replay a message with many different corruptions to determine it's content.
Yes, if you augmented this protocol with a digital signature it could accomplish authentication (if implemented correctly!); but since the protocol has no particular need for a digital signature otherwise the signature is an extremely slow way to accomplish authenticated encryption (compared to a proper authenticated encryption mode). An authenticated block cipher mode can be made pretty much just as fast as unauthenticated (at least for longer messages, and only (say) a 2x slow down for the shortest messages... one or more orders magnitude better than a digital signature).
It is very hard to design secure cryptographic protocols and systems and even harder to have a good basis to believe that they're secure. Whenever the primitives we use have subtle behaviour that differs from an idealized version that job becomes significantly harder. Authenticated encryption works a lot more like a simple black box "ideal encryption" than non-authenticated ciphers.
Secure systems could use either, but non-auth is a LOT easier to end up with an insecure protocol by surprise.