Adafruit CC3000 WiFi Module working together with MFRC522 RFID Module Issues both SPI over Arduino
I couldn't get your code to download so this might be wrong, but it sounds like the modules are running different SPI modes: http://en.wikipedia.org/wiki/Serial_Peripheral_Interface_Bus#Mode_numbers

I had this problem a few years ago. You'd expect SPI to be SPI, but it comes in four different flavours and they won't talk to each other on the same bus. Unfortunately there's no solution but to replace the modules; the Arduino can only run one SPI bus, so can only talk to one of these modules at a time. 

However you can run UART and I2C or UART and SPI concurrently by using SoftwareSerial (don't use the native Tx/Rx pins, bad things happen).

So your best bet is to switch to modules that use I2C or UART. Personally I'd use UART, simply for the simplicity of having two separate connections and a higher data rate.