The maths is identical — the inputs aren't
Both use the 4% rule: nest egg ≈ annual expenses × 25 (the inverse 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× multiplier, a difference in annual spending becomes a 25× 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 margin scales non-linearly. A lean budget has little room to absorb a market crash early in retirement (sequence-of-returns risk) 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 genuine optionality, which is partly what you're paying the extra years of work 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 against is materially higher than today's budget × 25. Plan the number in today's purchasing power, then inflate it to your retirement date — the FIRE and retirement-corpus calculators do this automatically.