What's the difference between bluetooth-soc chips and bluetooth modules?
The reason the CSR chip is referred to as an "audio platform", is that in addition to the Bluetooth capability, it also includes a DSP (Digital Signal Processor), stereo DAC and ADC, and a battery charger.  It is clearly intended to be used for products such as a wireless headset or headphone.

The BC127 module also has a DSP and DAC, but no ADC  (so it is intended for output only audio applications, like a wireless headphone).  It includes an antenna; the CSR does not.

Both are dual-mode: classic Bluetooth and BLE (Bluetooth Low Energy), and support many other protocols besides audio.

The CSR would be a bad choice for you by itself, as it comes in either a 112-ball BGA, or WLSCP package, neither of which are hobbyist friendly.  You would need to buy some sort of development board or breakout board; theirs is $800.

The BC127 is a better choice, but once again, the module by itself with castellated pins with a 0.8 mm pitch is not particularly easy to work with.

I suggest you look at SparkFun's offering of Bluetooth breakout boards; while you can buy the BC127 module by itself (cheaper than Digi-Key BTW), you are better off buying a breakout board which is easier to work with.