I have always used both "root" as in route 66 and "rooter" as in the networking device. The latter has gotten me funny looks often, however I could not bring myself to accept the inconsistency. Today I heard "rowt" used for a path of movement by a radio presenter. Which is correct?
In my local dialect (Toronto, Canada), it is /ruːt/ for a roadway, and /raʊt/ (but that's very approximate; see Canadian raising ) for the act of specifying a path (and rowter for the computer networking device)
It's a question of dialects. In the UK, it is pronounced as a homonym to root, as already been addressed. In America, it seems that those that pronounce it as a homonym to root are more concentrated on the east coast, as can be seen in a map (link below) visualizing the results of the Harvard Dialect Survey.
In a study of American dialects (link below), Stephanie Nicole Hedges finds that the probability to pronounce "route" as rhyming with "out" is 0.5 in New England, New York, and the Mid-Atlantic States, while it is 0.8 elsewhere in the USA.
US Midwest: I pronounce the noun 'root' when it's used to name a road ("Route 66"). If I am asking what roads you will take to get to Chicago, however, I ask, "What 'rowt' will you be taking?" And I might respond to a similar question by saying "The Triple-A 'rowted' me through La Crosse."
And the box that controls your internet traffic is a 'rowter'.
So 'root' (rhymes with "hoot") is used only as a "proper noun", and 'rowt' is used everywhere else.
(I suspect that the special treatment of named roads with "root" is due in large part to the TV show "Route 66" (1960-64), and the earlier (1946) song by the same name. This pronunciation permeated US culture in the 50s and 60s.)
Today, the pronunciation of route as /raut/ (rhyming with out) is more prevalent in the USA.
But the two pronunciations are not equally distributed in the USA. In a study of American dialects, Stephanie Nicole Hedges found that the probability to pronounce route as rhyming with out is 0.5 in New England, New York, and the Mid-Atlantic States, while it is 0.8 elsewhere in the USA. This is reflected in the map from the Harvard Dialect Survey given in the answer by Holly Rayl.
If you consider other factors such as people living in urban or rural areas, age, sex, and education the distribution becomes even more complex: women, older people, people in urban areas, and people with higher education tend to prefer the pronunciation of route as /ru:t/ (rhyming with boot), while men, teenagers, people in rural areas, and people with lower education tend to prefer the pronunciation of route as /raut/ (rhyming with *out), at least in Canada.
This may correspond to the finding that in Britain the pronunciation of route as /raut/ (rhyming with out) is the one common among the British army.
Historically, the pronunciation with a diphthong has been recorded since the second half of the 18th century and was at that time the preferred pronunciation of some, but not all of the commentators. It has disappeared from standard British English in the course of the 19th century.
Route, as in a way or course to be taken, is pronounced 'root' in non American English as it is in French from which it is taken. The verb, to send down a particular way, is the same. Hence the computing router which directs signals down particular ways, would be pronounced rooter. Except that the influence of Americanisms is so strong that I often hear 'rowter' here in NZ and on British TV programmes (not programs).
Rout (no 'e'), rhyming with about, means to force a disorderly retreat of beaten troops. It also means 'to force out as if by digging' which is where the power tool comes from. The words are NOT related and are pronounced differently.
Unfortunately this gives us one of the inconsistencies of English in that router (from route) would be pronounced 'rooter' and router (from rout) is 'rowter'. The Americans have put in a consistency in pronouncing everything rowt and rowter, non-Americans have maintained the original pronunciations and pronounce them differently with a spelling inconsistency.
Now, which is actually correct? Because I was brung up with Non-American English I would have to say root and rooter.