The first one means "it is not true that: either age is at least 65, or age is also at most 80". Or, put another way (
see DeMorgan's laws on Wikipedia): "age is not at least 65,
and age is not at most 80". We can simplify this noting that the inverse of "at least" is "less than", etc.: "age is less than 65, but more than 80". This is a contradiction.
The second one is equivalent to "age is less than 65 or more than 80", which is the opposite of what you want (this is true only when age is not in the range you want).
Just build it by using English. "Age is between 65 and 80 inclusive," becomes "age is at least 65
and age is at most 80". Does that help...?