Sup is the supremum and max is the maximum. A bounded sequence must have an supremum, but it may not have a maximum. For example, an= 1- 1/n, this sequence has supremum = 1, but it can't get the maximum value. But if a sequence can get the maximum value, it means that the maximum value max exists.

Similarly, a bounded function on an interval must have an supremum, or it may not have a maximum. Similarly, for example, y= 1- 1/x, this function has an supremum on [1, positive infinity], but it can't find a maximum value anyway. Similarly, this function can't get a maximum.