This is one of the few usage rules where American and British English genuinely diverge, rather than one having drifted from the other.
Used for physical, measurable distance.
Used for figurative distance, degree, or the sense of "additional." Also a verb meaning to advance something.
Quick way to remember
If you could measure it in miles or metres, use "farther" — both contain "far." For anything abstract, additional or advancing, use "further." When you genuinely cannot tell which, "further" is the safer choice; it covers both in British usage and reads as natural in American.