So the info states that "walking or running does not affect detection", That means I can run around all I want and still be as invisible as while sneaking?

Ie footsteps make no noise? Ie I don't need Muffle on my boots anymore?

I don't like how some perks, items, or spells completely and unequivocally obsolete others. There ought to be some tradeoff.

How does that answer my question?

Sorry, I have the bad habit of running off with a topic and letting it go far away in some forgotten corn field. If I could rectify my mistake, I would.

I just watched Pulp Fiction, fo the first time in over a decade, and Vince said that by asking for forgiveness automatically makes you forgiven

A monster can hear your footsteps when you're not sneaking. Does that mean anything to you? For me it means you have a few more fractions of a second to crouch before they notice someone is lurking in the shadows. The usefulness of this perk is arguable though and for those who have problems sneaking this is where illusion shines.

Well by reading the perk-info it states that it does not matter if I run or walk. Since I cannot run while sneaking I have the understanding that I am as stealthy while running and walking normally as if I'm sneaking.

I agree with you.

Well, as i see it while sneaking you can decide whether to approach carefully (walking by pressing shift) or simply moving a bit quicker as it does by default while stealthed. In my understanding i would say that perk removes the difference between those two approaches, in another words: it doesn't matter whether you go slow or fast at your target, provided you are sneaking of course. I hope it helped.

It means that you can move around all you want while sneaking, be it slowly or quickly, and it won't affect your detection.