Though both are the closest synonyms of each other, there might be a slight difference between the two, but only in application. "Calculate" is the more commonly used synonym, and "compute" possibly began to be used more ever since computers became widespread. The meaning of both the words overlaps, as compute is essentially "to calculate (or reckon) an amount or figure; or to make a calculation using a computer, etc." Note the word "calculate" is included in the meaning of "compute". Compute is used more in sentences which are related to tasks related to usage of computers (which could be the only possible difference in the application of both the words).