First statment: This statement seems to be an easy fall statement. At first it seems fairly obvious that if on increasing a value by 1, the number of distinct integers do not change. But that would be true only if all the numbers are distinct. Here it is clearly mentioned, NOT necessarily distinct. For example if the list has following numbers 1,1,2,5,... rest all distinct. If the number 2 is increased by 1, still the number number of distintct integers are same, but not consecutive. NOT SUFFICIENT
The second statment at first may seem without clue. But it turns that this is a similar case to the previous it is NOT SUFFICIENT to answer the question.
Now let us test the answer for both statements combined. The key is here think of set, dont do it your brain. Don't tax your brain.
Suppose the number set we choose is 7, 7, 8, 10, etc.
Now if you increase 5 by 1, you get 6 and number of distinct integers stays the same! In this case, if there are no consecutive integers, the number of distinct integers will increase. Hence if the numbers are not all distinct and the number of distinct numbers needs to stay same, there must be a pair of consecutive integers.
Both statement combined are sufficient.