Encrypted data should be indistinguishable from random noise, whereas there is definitely order to plaintext compressed data.
A good compression method also maximizes entropy (by compressing the same message into a much shorter space)
If there is still some structure (non-randomness) in the message, then you could use that knowledge to compress the data further. (shortening the message at the cost of a more complex decompression method).
Therefore a "perfect" compression should look like a header + random noise, just like an encrypted message.
At the limit of compression, with known predefined probabilities of messages being passed, there is little difference between the compression and encryption - "message 1 = i want to have lunch with you tomorrow", message 2 "i agree", message 3 "how about the day after" Is both a great compression scheme for a specific type of communication, and also rudimentary encryption
Alice:1
Bob:3
Alice:3
Bob:2
(and then they have lunch 3 days from now)