I remember from my cigarette days that if I let an opened pack
Sit around for a week or so and then lit one up the taste would be horrible. Stale. But while I hear about cigars being over humidified or dried out or whatever I have never heard one referred to as stale. Why is that?

Well, if you leave a cigar "out" in the elements for a week, depending on the condition of the elements, it too will suffer. What one wants to call it doesn't matter, stale, dried out, over humidified...............it's simply not in good smoking order.
A cigar, typically being much larger than a cig, will hold up longer in less than ideal conditions, and doesn't have all the other crap in it that a cig has.
Finally, I'd wager that if you kept an open pack of cigs in a humidor, they wouldn't go "stale" nearly as fast..........but I'm not sure about the optimum temp and rh for cig baccy.