I remember learning that the difference between opening 4NT and 5NT was that 4NT invites 6NT, 5NT forces 6NT and invites 7NT.
I opened 5NT once, and only once, and my partner passed 
Umm… 4NT and 5NT are special responses to partner opening 1NT or 2NT, with meanings as you describe.
Just 3 weeks ago I responded 5NT to an opening 2 NT bid and my partner passed. 
I don’t play a lot of bridge, but I’d be confused if my partner opened 5NT. I would have passed, too, not knowing what you meant by 5NT. Opening bids, as far as I know them, are conventionally always at the 1- and 2- level, aren’t they?
Why not bid 4C (Stayman) if you’re interested in slam?
Oh, you’re exactly right.
3C would be Stayman, not 4C. An immediate 4C over 1NT or 2NT is commonly ace-asking (Gerber).
An immediate bid of 5NT in response to partner’s 1NT or 2NT opening is usually “pick a slam” which gives partner a chance to suggest a suit at the 6-level, or bid 6NT. Hence with a hand like AQX KQx Ax KQTxx and partner bids 5NT over your 2NT opening, you bid 6C to suggest clubs.
Bidding Stayman over partner’s opener will not help you find a minor suit slam.
You’re right. Been so long since I’ve played I misremembered.