From the FAQ in Firecalc:
How does FIRECalc compare to "Monte Carlo" simulations I have read about?
Both FIRECalc and Monte Carlo simulations are trying to get away from using a single historical or assumed average rate of return and inflation rate. Neither can predict the future, so both have to come up with some means of putting together numerous potential future scenarios. In effect, both programs are rolling dice to come up with scenarios that presumably could happen in the future. Such a simulation needs many sample situations -- the more the better, as long as they are realistic -- to give meaningful answers.
One approach to this, which is what a Monte Carlo simulation does, is to define a lot of rules about how much inflation might happen, how the markets might perform, and so forth. Expert analysis is used to insure that the range of such values is realistic. Then they roll the dice many times to create many possible future sets of market conditions, inflation, and so forth, all applied to your portfolio and withdrawal plans. They are using their theory about what is possible, and then rolling the dice to see how different combinations of years might impact you. Presumably they are limiting the possible market returns in a single year to some range that they believe is within reason.They would also limit inflation to some range, and probably tie interest rates to inflation somehow. Using these hypothetical situations allows them to "roll the dice" many hundreds of times for each portfolio. To the extent that their rules about possible future market conditions are reasonable, their approach is certainly sound.
In FIRECalc, and in the REHP spreadsheets on which FIRECalc was based, we use no theory, and certainly don't claim to know enough to say what might happen. Since we're looking at safety and not optimizing portfolio balance or income, the task seems simple. Would a given strategy survive the worst that has ever happened in the US, or not?
With FIRECalc, you're accepting that the next 30 years or so won't be any worse than the worst we've seen in our actual history. With a Monte Carlo simulation, you're accepting their definitions about what is possible in the future.