I love using potato ricers, and I was truly amazed at how perfectly you can make mashed potatoes with them, without any lumps and without any worry of over mixing. Prior to getting one, I would most often make "whipped" potatoes with my electric hand mixer. However I always had to be very careful with that method since it was really easy to over mix the potatoes which would result in really gummy gluey mashed potatoes. Also, if the potatoes weren't cooked to the right tenderness, no matter how long you mixed it, they would not break down entirely, so it becomes even worse - they're gluey but also lumpy.
The potato masher was fine if I was making more rustic mashed potatoes. I would usually use that if I were making mashed potatoes with the skin on, since I wouldn't have to worry about the thick skins perhaps damaging the mixer blades or getting caught in them and slowing the mixer down. I actually don't prefer the wire one pictured, but rather the kind that looks more like a grid design with square openings. The one pictured above takes much longer to mash everything.
Some potato ricers are so well designed that you can actually just place a whole potato in them, skin on, and it will extract the filling and leave the skin in the ricer. My sister has one that can do this, it's an all stainless steel model with relatively tiny circular holes. My folks bought me the Oxo one for Christmas one year, which is adjustable so you can choose how coarse or fine you want the ricer to be, but honestly, I prefer my sister's better. The Oxo one will often let stuff like the skin through it, and it's more of a hassle to clean out because the skin gets stuck in it.