Nymphing & Stealth

[Eagle Claw]

Here is a question, I am not a very good nympher, however I am somewhat confused when it comes to stealth and nymphing. When dry fly fishing a big deal is made of stealth, sneaking up on fish, not causing any vibrations, noise or shadows for fear the fish will scatter or be put down, especially wild trout. Yet with nymphing, people almost step on the fish by wading right up to where they are and drifting a nymph a rods lengt away ,so whats the deal with stealth ? How can you be sneaky and stealthy when walking right up onto the fish ? Stealth seems to contradict the methology with dry and nymphing tactics.