The words ‘root’ and ‘route’ are homophones in British English (in other words, they are pronounced in the same way), whilst most Americans would pronounce ‘route’ to rhyme with ‘out’ or ‘shout’.
The root is that part of a plant which grows downward into the soil, anchoring it and absorbing water and food. This original meaning has proved metaphorically fertile, and the word has diverse figurative usages, not least in etymology, where the ‘root’ of a word is its base semantic unit (for example, ‘sem-‘); it is also used in technical fields such as mathematics, computing and genealogy, but always with a similar figurative application. And it is a common figure of speech to express the base or the origin of something: ‘the root of the problem is…’; ‘the root of his unhappiness…’
A route (as a noun) is a way or road or direction. Derived from the French ‘route’, road, it is associated with the verb ‘to route’, meaning to give a particular direction to something: ‘the policeman routed the traffic down the sidestreet’. It is also the source of the name of that little box which by some ineffable and arcane e-magic fills your home with internet, namely the router, so called because it ‘routes’ whatever strange forces are needed to generate wifi.
In English the phrase ‘en route’ is sometimes used to mean ‘on the way’. It is always spelled as it is in French, from which language it was borrowed.
A related word is ‘rout’, pronounced (in both British and American English) to rhyme with ‘out’ or ‘shout’. Although the OED lists twenty meanings (as both noun and verb) of the word, the most important one is ‘a disorganised retreat’, when an army or other unit panics, breaks and flees willy-nilly from its adversaries (who, it is imagined, give chase and, to coin one of those brilliantly disingenuous military euphemisms, ‘mop up’.)
It is in this sense that it is most often used metaphorically: a football team, for example, can be routed, after the tenth goal is scored against them, or a politician can rout his opponent at the ballot box.
Curiously, both route and rout share a common root: the Latin ‘ruptus’, broken.