The mode of a set of integers is x. what is the difference

I guess someone already pointed out that Statement 1 does not tell you your numbers are consecutive multiples of 3. It says that no two numbers are more than 2 apart. And in your comment following the 'HOWEVER', if every element in a set occurs exactly once, then the set is said to have no mode at all. For a set to have a mode, some element needs to appear more often than at least one other element. So sets like {1, 3, 5} and {3, 3, 4, 4} have no mode, because they do not contain any element which appears more often than some other element in the set.

In this question, neither statement is sufficient alone. For Statement 1, our set could be {1, 2, 2, 3}, and then the median and mode are equal, or it could be {1, 2, 3, 3}, and our median and mode are different. For Statement 2, again our set could be {1, 2, 2, 3}, and our median and mode can be equal, but our set could be {1, 2, 3, 4, 4, 10}, and our median and mode are different.

Now, using both statements together, if the mean is equal to the mode, then the mean must be equal to some value in the set. Technically, if all the values were the same, there'd be no mode, and we cannot have only two values exactly 1 apart, because then the mean would not be an integer, and thus would not be in the set. So we must have at least two values which are exactly 2 apart. Let's call them s-1 and s+1. So, our set has some elements equal to s-1, possibly some elements equal to s, and some elements equal to s+1, for some integer s. Notice now that the mean needs to be in the set, so must be s-1, s or s+1. But the mean can't be s-1, since s-1 is the smallest element in the set, and we have elements larger than s-1 in the set. Similarly the mean can't be s+1. So the mean, and therefore the mode, need to be equal to s. And for the mean to be equal to s, the number of elements equal to s-1 must be equal to the number of elements equal to s+1. So our set must be symmetric, and the median must also be s. So the median and mode are the same, and the two Statements together are sufficient.

edit: I'd add that I don't think I've ever seen a real GMAT question that even mentions the mode, let alone one as tedious to solve as this question, so it probably isn't all that important to study.