Your basic phrasing of the question is internally inconsistent: two identical set of data A and B means that there are no differences. If you mean similar data structure, number, name and types of variables then state so.
By "largest value" do you mean a single difference for exactly one variable in one set with all other values for all variables the same? Then the answer about median can't be answered without exact data. If the single value replaced by the largest is the largest (not stated, you only say the largest is 3 times greater not that it replaced the other) for the corresponding variable in the data there would not be any change in median (order of the value is important), if some other value is replaced then there could be a difference in medians based on your specific rules for tie breaking.
Replacing a single value would effect none of the other variables at all.
If your data set is "large", which might depend on the actual range of values in the sets, the difference in mean or standard deviation might not be easily detectable.
Suggestion: Take a data set and modify a copy then summarize the two sets. Repeat with data sets with many more records.
