Why this is harder to do by hand than it seems
The obvious approach — count the months between two dates and multiply by roughly 30 — is only ever an approximation, because calendar months range from 28 to 31 days, and that variation compounds the more months are in the span. Add a leap year's extra day into the range, and a purely mental calculation can easily be off by several days, which matters a lot if the number is being used for anything precise, like a legal deadline or a billing period.
The exact method
The reliable way to get an exact day count is to convert both dates to a single continuous count — days since a fixed reference point — and subtract. This is exactly what happens under the hood in spreadsheet software and date libraries: every calendar date maps to a serial day number, and the difference between two dates' serial numbers is the exact day count, automatically correct regardless of which specific months or leap years fall in between.
Calculate the exact difference
When you need more than just a day count
A single number of days is useful for some purposes (a countdown, a billing period), but a broken-down years/months/days figure is more useful for others (describing someone's age, or how long a relationship or tenure has lasted, in a way that reads naturally). Getting that breakdown right requires a slightly different calculation — comparing the day, month, and year components directly and borrowing across units when a component goes negative, similar to manual subtraction with regrouping — rather than just converting to a single day count.
Common situations where this comes up
Counting down to a deadline, wedding, or trip; calculating exact tenure or time-in-role for HR purposes; verifying a contract, warranty, or lease's remaining days; and checking a billing or interest calculation that depends on the precise number of days in a period rather than a rounded month count are all common, everyday reasons to need an exact date difference rather than a mental estimate.