The traditional rule
The classic distinction is straightforward: use "farther" for measurable physical distance and "further" for abstract extent or degree. "The airport is farther than I thought" (physical distance). "We need to discuss this further" (more discussion, not physical movement).
This rule has been taught in style guides for over a century, and it remains the safest approach for formal and edited writing.
Where the overlap happens
In practice, "further" has been encroaching on "farther" territory for decades. Most style guides now acknowledge that "further" is acceptable in both physical and abstract contexts, while "farther" is limited to physical distance. You can say "further down the road" without anyone objecting, but "farther into the discussion" sounds wrong to careful readers.
Think of it this way: "further" is the flexible one and "farther" is the specialist. When in doubt, "further" is the safer bet, unless distance is clearly measurable, in which case "farther" is more precise.
- The store is farther from here than I remembered. (physical distance)
- Let me explain this point further. (additional degree)
- We drove further into the desert. (acceptable, though purists prefer farther)
- Nothing could be further from the truth. (abstract, only "further" works here)
"Furthermore" and "further" as a verb
"Furthermore" always uses "further", there is no word "farthermore." Similarly, when "further" is used as a verb meaning "to advance or promote," there is no "farther" equivalent. "She worked to further her career" is the only option.
These cases are useful anchors. If you remember that "further" is the word with broader grammatical range, the comparison cases become easier to navigate.