I absolutely recommend fenders.

I had been commuting on my road bike without fenders for a while (couple of months). The worst part is the backsplash of water and mud on your back, and on the top of your butt. When wearing a backpack, it gets absolutely sprayed in mud every time too.

I finally had enough of it, and got some Portland Design Works fenders. They are expensive, but nice-looking, and they do the job very well. I now arrive at work or home with a non-muddy back, and my backpack stays clean. My bike stays clean. I love it! I am far less bothered by rainy rides in general now.

Apart from keeping yourself clean, it does keep the bike A LOT cleaner as well. Without fenders, the mud and water is sprayed all over your BB, Cranks, chainrings, chain, seat tube,, etc. I don't think it does damage short term, but long term all that sand and mud inside of those moving parts will wear them out more quickly. So yes, even from a mechanical point of view, it is a good idea to have fenders. Your bike will last longer / require less maintenance.

Fenders exist in all price ranges. Why would you NOT get fenders? If they really annoy you, there are some that are "easily" removeable, so you can remove them on sunny days. I don't have fenders on my nice road bike, but I only ride that one in fair weather.