If you are a "fair weather" cyclist, you don't need fenders, but if you are a serious cyclist, and don't live in a desert climate, you really should have at least one bicycle with fenders.
Fenders make a tremendous difference when you are riding roads that are wet from drizzle, recent rain, or snowmelt.
In a hard rain, you may become wet with clean rain from above, but fenders will protect your body and bicycle from the mud and sand kicked up from dirty puddles and rivulets.

Fenders Protect Your Bicycle

Many cyclists protect themselves from rain by wearing rainproof clothing, but they forget that their bicycles don't like dirty water any better than their bodies do.
The water kicked up by your wheels is much worse for your bicycle than the clean rain falling from the sky. If you ride in wet conditions without fenders, your chain, derailers and brakes will all get sprayed with sandy, muddy, scummy water, often mixed with gasoline residue. This is very bad for these parts.
Even more vulnerable is the lower section of your headset. Headsets are designed to shed water like the shingles of a roof, and are basically rainproof...but the gritty spray from below has easy entry to the bearing surfaces of the heavily-loaded lower races.

Fenders Keep mud from flinging up your back

Have you ever ridden your bike in the mud, and showed up back at home with a giant line of mud up your back?
Even worse, have you ridden your bike in the mud, then went to a restaurant afterwards, knowing everyone else there doesn’t understand why there is brown stuff flung up your backside?
Yeah, rear fenders are going to help you with this! They stop mud from being flung up from the rear tire and onto you.
This reason alone should convince you to get fenders on your bike.