A4 has 1.0339 times the area of US Letter. The two are close enough that a document printed for one fits awkwardly on the other, and different enough that it always shows — A4 is 18 mm taller and 6 mm narrower.
About the units
A4 is 210 × 297 mm, part of the ISO 216 series adopted from a German standard of 1922 and now used by every country except the United States, Canada and a few others. US Letter is 8.5 × 11 inches, or 215.9 × 279.4 mm, and its origins are undocumented — the most cited explanation, that it derives from the size of a vat of pulp a papermaker could handle, has no firm evidence behind it. The ISO series is defined by a rule; Letter is defined by tradition, which is the fundamental difference between them.
The exact factor
The area ratio is 62,370 mm² for A4 against 60,322 mm² for Letter, giving 1.0339. But the shapes differ too, and that matters more in practice than the area: A4 has an aspect ratio of 1.414 while Letter is 1.294, so A4 is proportionally longer and thinner. Scaling A4 content to fit Letter requires 94 per cent reduction to preserve the height, which leaves visible side margins; scaling Letter to A4 at 97 per cent leaves a gap at the bottom. Neither fits without either cropping or white space.
Where you meet this conversion
International document exchange, which is where the mismatch causes daily friction. A PDF laid out for A4 printed on Letter loses the bottom of the page or shrinks; word processors default to one or the other based on locale, and a shared template silently reformats. Business cards, envelopes and filing systems all inherit the split. Printer drivers offer a fit-to-page option specifically because of this pair, and the resulting scaling is why a document's fonts sometimes look subtly wrong after crossing the Atlantic.