Why does this 5 V power supply work and the other one doesn't?
The problem is probably interference between the power supply and the telephone line.

Telephone lines and power lines should be isolated from one another, but cheap devices (the DTMF decoder, or the power supply, or both) may skimp on isolation - if it were a circuit with a speaker, you would hear a loud humming noise.

This goes along with the USB connection on your computer.  It works until you plug in the printer.  The printer can also cause interference.

It might be as simple as a"noisy" power supply (the 5V is more like "5V with garbage on it") or it might be that the isolation is poor and you get what are called "ground loops."

In any case, use which ever power supply works best.



That module is (according to the seller's website and the datasheet of the IC on the module) not a caller-ID module.  It is a DTMF decoder with a 4 bit binary output.