Based on what I know, symbolic computing is based on hard rules in the code. Statistical translation is based on statistic and probabilities. If so, could you give an easy to understand example to demonstrate this difference? If it is not correct, please give a correct explanation of the differences.
By symbolic translation, I assume that you meant rule-based translation. As a simple example, consider translating the English sentence "I eat apples" to Spanish.
Rule-based translation works by first tagging the words with their part of speech ("I" as noun/subject, "eat" as verb, "apples" as noun/object). Then, create an English parse tree and transform that into a Spanish parse tree (this requires knowledge of how English phrases map to Spanish phrases). Lastly, translate each English word into its corresponding Spanish word, and use the Spanish parse tree to determine the structure of the sentence.
The simplest statistical machine translation system are n-gram models based on n-gram frequencies. For an explanation of the mathematics behind statistical translation, see this paper: http://www.mitpressjournals.org/doi/pdf/10.1162/coli.2006.32.4.527.