Some authorities offer differentiated meanings for farther and further, but the short answer to the question of which to use for what is that you can just take your choice.
The word farther is a comparative of far. Your house is farther from the school than ours.
The word further comes from an Old English word meaning “to impel.” We still use the word further as a verb: He gives frequent speeches in order to further his cause
As adverbs, both have come to mean “at a greater distance.”
How much farther is it to the station?
I’m too tired to walk any further.
The OED says
In standard English the form farther is usually preferred where the word is intended to be the comparative of far, while further is used where the notion of far is altogether absent
It concedes, however, that “there is a large intermediate class of instances in which the choice between the two forms is arbitrary.”
According to the Online Etymology Dictionary,
There is no historical basis for the notion that farther is of physical distance and further of degree or quality.
As long ago as 1926 H.W. Fowler (A Dictionary of Modern English Usage) observed:
The fact is surely that hardly anyone uses the two words for different occasions; most people prefer one or the other for all purposes, and the preference of the majority is for further.
Differentiating between farther and further as adverbs could be useful. We could use farther only when actual distance is involved and save further for other uses. But as Fowler observed, most people are not going to use the words in that way.
About all one can say is that, as adverbs, farther and further are interchangeable. As a verb, further is the preferable form.