I am trying to better understand the Heston model and its implementation. It seems like a lot of people use the FFT method for calculating the call prices during the Heston calibration, but the Monte Carlo method is used to calculate the prices with the calibrated parameters. What is the point in this? Why not just use the FFT method for calculating both prices?
The big advantage of the Heston (1993) model is that it admits a quasi-analytical formula for the pricing of European options. So, obviously, if you're going to price European options, you should be using some kind of numerical method to approximation the integrals in the quasi-analytical formula. The exact method you use will depend on the case at hand and what matters most -- usually, speed is of the essence.
However, that only works for European options. Outside of that small pond, it might be possible in some cases to think that your European option is a good starting point for calibration. I have never done this personally, but I have done similar things (use simpler methods to reduce my searching time before doing things in a more Kosher manner).