A well-known difference between mean and median is that the latter is more robust. But there are a few more differences (they are of course not all equally important and their importance depends strongly on the context: what is crucial in one case can be utterly irrelevant in another).
1. The median of a set (with an odd number of elements) is a member of that set, not so with the mean. For instance the median of integers is an integer whereas the mean may be fractional. The latter is especially problematic when mapping something intrinsically discrete onto integers. With data on revenue day per day a company can find out which day of the week is the mode and which is the median; but without time information on top of day, a mean does not signify much: if Wednesday is 3 and Thursday 4, it is easier to calculate an average of 3.4 than to understand what it means. Median wins.
2. All it takes to calculate a mean is two numbers: the total and the number of contributions. The median requires the whole distribution, which is not always so readily available. If there are five million cars and four million households in a country, what is the median number of cars per household? Go fish. Mean wins.
2′. And from the mean and the number of contributions one can calculate back the total. If the median size of the cattle of ten farmers in a certain village is 30 cows, what is the total number of cows? Mean wins.
3. Unlike the mean the median only requires ordering. If you are given (an odd number of) gray cards and are asked which is the median shade of gray, all you have to do is order them from darker to lighter and pick the one in the middle. But in order to find the mean it would be necessary to quantify the shades of gray, which is less straightforward and less objective. All the median needs is the ability to tell which of two shades is darker, not by how much. Median wins.
4. Moreover, a mean can be easily calculated in certain situations where the median cannot even be conceived. Per capita gross domestic product (GDP) is calculated as the GDP of a country divided by its population, i.e. it is an arithmetic mean. But what would be median GDP? It would require to know the distribution of GDP person per person, which makes no sense. Likewise with density of population (number of inhabitants over surface area of the country): this is implicitly a mean and there is no median equivalent (it would require to assign all space to a different person to get a distribution). Mean wins.
5. The mean is more likely to change if functions are applied. For instance, with an annual return of r an investment will return (1+r)n − 1 over n years. Since the function turning annual into total return increases strictly, the median annual gain is the same as the median total gain. This is not true with the mean: annualized mean gain is not the same as mean annualized gain (only affine functions conserve the mean). Median wins.
6. The arithmetic mean is always the sum over the number of contributions, even in the multivariate case. But, in that case, defining (and computing) the median is harder and more ambiguous. Mean wins.