I prefer to replace tubes on the road and patch tubes at home. That said, it's worth carrying a patch kit (see any LBS) in case you get two flats on one ride. And remember to put a fresh tube in your bag when you get home after replacing a flatted tube.
I'm using up my second box of
Rema patches, still have about 80 to go (with luck, another 10 years' worth). After watching two bottles of (expensive) Rema fluid polymerize before I used it up, I'm using Slime cement from my local auto parts store at half the price. Despite the nay-sayers, I've found Elmer's rubber cement to work well, too.
The keys to making a patch work are, first, sand or grate the area around the leak well, enough to take the shiny mold release off the outside of the tube. Second, when the tube surface looks dull, put a thin layer of cement on and let it sit until it dries (hard to do when you're outside in the rain!). Then peel the aluminum foil off the back of the patch and press it on firmly.
One more tip: get a silver Sharpie and mark the location of the puncture. If you patch in batches of, say, half a dozen at a time, you'll save a lot of aggravation if you blow up the tube when you get home and run it through a sink full of water looking for bubbles. Mark it then, then save it for company. When patch day comes, sand, glue, sand the next tube, glue the next one, patch the first tube, and continue like a one-person assembly line.
Finally, on to the "why patch." Economics, for me. I don't remember buying new tubes for less than $4, and my last purchase was @$8. I can patch them for $0.25-0.60, so (worse case) I spent $60 to save $400. Yes, it takes time, but nobody pays me for sitting around the house on a rainy afternoon. There are those who refuse to patch; that's their choice. I've had almost as many flats from poorly made tubes right out of the box as I've had patches fail.