Unlike cushioned running shoes and their squishy brethren, weightlifting shoes, which feature a slightly raised, wood heel, give you a stable base for lifting. The heel doesn’t compress or wobble on you, the footing is flat and solid, the heel is slightly raised, and they efficiently transfer force between you and the floor, which is what lifting is all about.
They’re definitely worth having if you plan to squat, deadlift, press, clean, jerk, or snatch anything more than a dowel.
You can't buy them in your local shoe store, but Todd Lyons will ship them to you virtually overnight: Adidas weightlifting shoes. You need to get some!