The maths is the same. The inputs aren't
Both use the 4% rule: savings target ≈ annual expenses × 25 (the flip side of a 4% safe withdrawal rate). Lean FIRE might plug in a frugal annual budget. Fat FIRE plugs in a comfortable one. Because of the 25 times multiplier, a difference in annual spending becomes a 25 times difference in the target. And that's before inflation lifts both numbers over the years until you retire.
Why the gap is bigger than it looks
Fat FIRE isn't just 'more'. The safety cushion grows faster than the budget. A lean budget has little room to absorb a market crash early in retirement (the risk of bad returns in the first years) or a healthcare shock. So many lean retirees end up needing a side income (Barista FIRE) or a lower withdrawal rate (3.5%), which pushes the real target up again. Fat FIRE buys real freedom of choice, which is part of what the extra years of work pay for.
Inflation: the input people forget
Your expenses today are not your expenses at retirement. At 3% inflation, costs roughly double every 24 years, so the real target you should plan for is well above today's budget × 25. Plan the number in today's buying power, then grow it to your retirement date. The FIRE and retirement savings calculators do this for you automatically.