Maximum amplitude of the signal could be defined as max(signal), depending on what is meant.
However, range of the signal is the difference between the maximum and minimum of the signal. Sometimes when "maximum amplitude" is being referred to, it is the range that is being discussed.
For example, if you had a signal that varied between 0 and 1, then the range is (1-0) = 1, and the maximum would be 1. Now if you have a signal that varies between -1/2 and 1/2 then the range is (1/2 - (-1/2)) = 1, and the maximum would be 1/2. Sometimes that distinction between the peak at 1 and the peak at 1/2 is important and sometimes it is not important.
