Does the Silent Running perk mean that I don't have to be crouched in order to sneak?

There are four modes of movement:

Run, Walk, Sneak Run, Sneak Walk.

Most people don't know about Sneak Walk, but it is the slowest speed. If you are sneaking and can catch up to a walking person, you are doing Sneak Running.

You are more easily detected when Sneak Running than Sneak Walking. The perk allows you to Sneak Run without being more easily detected.

Not quite. Your speed is no longer a factor, but in order to perform the same acts as when you are sneaking (ie. Pickpocketing, etc.) you still need to be crouched.

So technically speaking, you aren't always sneaking. It behaves the same as before, but now you don't have to be cautious about your movement speed while crouched.

even the wiki doesn't have a helpful description, but this comment on gamefaqs describes it pretty well.

When you're crouched, there's still two different speeds. I think holding shift puts you into the "walking" state. This is who you move silently while crouched without silent running. If you don't, and just hold forward, you're normally auto-running, so while crouched, you're... I don't know, uh, running is a bad description.. roadie running?

Not sure what you're playing on, but shift on a PC is analogous to holding the analog stick part of the way instead of fully.