Why would we suddenly want to change the definition of distance? (Euclidean geometry has been doing fine in the last 2000 years...)
There are a few possible answers to this question. The most obvious one is suggested by the name of taxicab geometry. Euclidean geometry measures distance "as the crow flies", but this rarely constitutes a good model for real-life situations, particularly in cities, where one is only concerned with the distance their car will need to travel, and cars certainly don't fly (yet).

As a less serious application, taxicab distance is the right model of distance for some games played on a square grid and where only vertical and horizontal moves are allowed. (Arguably, most such games actually allow diagonal moves, in which case we have to use yet another definition of distance.)

Another very good reason for studying taxicab geometry is that it is a simple non-Euclidean geometry. As Eugene F. Krause writes in the introduction of his book (see bibliography), "To fully appreciate Euclidean geometry, one needs to have some contact with a non-Euclidean geometry." Taxicab geometry has the advantage of being fairly intuitive compared to some other non-euclidean geometries, and it requires less mathematical background. This is why the manhattan metric can also give rise to all kinds of fun stuff in recreational mathematics. Martin Gardner gives a few examples in the Scientific American.