Mean (at least arithmetic mean) strictly means adding the numbers then dividing by how many numbers there are.
Average could be the mean, but it could also be the median if there are one or two values that distort the mean. Example: what does the average worker earn if these are the salaries: 20000, 20000, 20000, 21000, 21000, 21000, 300000.
And average can also be the mode, especially when the data are not numerical "the average person will prefer a red car".