Difference between 'I need a minute' and 'Just a minute'?

Is there any semantic difference between "I need a minute" and "Just a minute"? I have no context.

I guess that the second phrase is a request of waiting to a few minutes: 1, 2 or many minutes. It's an unofficial style.

'I need a minute' has the more exact length of time. It isn't 2 or 3 minutes. It is punctually, official exact one minute!

Am I right?

Update1. I completed a little survey... So, I found out that 'I need a minute' is very exact. I have fear to post references to different sites for evidence (can be off-top).

Update2. Just a minute (idiom) = wait a minute. In other words. This imperative mood can't be polite.

"I need a minute" explicitly indicates that the speaker is the cause of the delay. "Just a minute" does nothing to indicate the reason for the delay. It simply indicates that there will be one.

"Just a minute" is a bit more formal than "I need a minute" which might be sharing more than I care to know.

Both phrases are polite if said nicely.

If taken literally, the length of time is 60 seconds for both but if I started timing someone who said this they'd likely be annoyed. They are used similarly to "just a sec" which is literally 1 second yet hardly ever is. The phrase that saves you from the pedantic people here is, "just a moment", which is vague enough to be literally true without involving a stop watch.