The most common definition of root is the underground portion of a plant (though this sense is usually used metaphorically). It has many other meanings, however, including (1) to dig with the snout or nose, (2) to rummage, and (3) to give audible encouragement for a contestant or team.
The meaning of route is narrower. Route refers to (1) a road, course, or way from one place to another, (2) a customary line of travel, (3) a means of reaching a goal, (4) a fixed course for a salesperson or delivery person, or (5) to send on a route. The word is also used in the French loan phrase en route (sometimes spelled enroute), meaning on or along the way.
Depending on the speaker’s background and the sense being used, route is variously pronounced root and route. In the former pronunciation, it is homophonous with root. In the latter, it is homophonous with a third word, rout, which also has several definitions, including (1) a disorderly retreat, (2) an overwhelming defeat, and (3) to defeat overwhelmingly. Also, somewhat confusingly, it shares with root the senses to dig with the snout and to rummage.
Gardening took root in the curriculum as a way to help the students build self-confidence and a sense of service. [Albany Times Union]
Hogs work the compost when they root around in their winter piles of straw looking for cracked corn. [The Union of Grass Valley]
This is not simply a guide to which team is going to win, but also whom you should root for. [Denver Post]
A large tree fell across Route 3A Thursday morning. [The Patriot Ledger]
A veteran MBTA bus driver accused of driving his route while drunk has been fired. [Boston Globe]
It was my job to compile the full report for each sub-launched missile and route it to a list of recipients. [Yahoo! News]
“I’ve become immune to it. It’s part of the business,” said Iguodala, who scored two points in a little more than 11 minutes during the rout of Argentina. [Denver Post]
Warwickshire’s extraordinary routing of their Midlands rivals Worcestershire before lunch on the opening day will serve to kick the recent habit. [Guardian]
Fairfax Media has suffered its biggest single-day share rout since early 2009 after writing almost $3 billion off the value of a suite of assets. [Herald Sun]