"firmata" is a sketch/protocol/library that allows a program on your PC to access hardware on the Arduino; it's something that links PC programs to external hardware connected to the arduino.
the IDE is the PC program that edits, compiles, and uploads programs (sketches) to your arduino.
So for example, you could have a "Processing" sketch that reads dials and sliders connected to the arduino, and uses their positions to modify a graphics display. To run all this, you'd use the arduino IDE to compile and upload Firmata to your arduino, and then you'd use Processing and it's Firmata libraries to compile and run the PC-side application, and the PC and Arduino would end up talking to each other...