Target question: Is d ≠ e ?
This is a good candidate for rephrasing the target question.
If d = e, then we can say that d - e = 0
Likewise, if d ≠ e then we can say that d - e ≠ 0

REPHRASED target question: Is d - e ≠ 0 ?

Statement 1: d > e + 1
Subtract e from both sides to get: d - e > 1
If d - e is GREATER THAN 1, we can be certain that d - e will NEVER EQUAL ZERO.
In other words, d - e ≠ 0
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT

Statement 2: d < e + 2
Subtract e from both sides to get: d - e < 2
If d - e is LESS THAN 2, it's POSSIBLE that d - e = 0, and it's POSSIBLE that d - e ≠ 0
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

ASIDE: If you're not convinced that statement 2 is not sufficient, consider these two cases that yield contradictory answers to the target question:
Case a: d = 3 and e = 2. In this case, d - e ≠ 0
Case b: d = 3 and e = 3. In this case, d - e = 0

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Dear GMATInsight,

Why did you conclude that d>1.

If d=-2 & e=-4.............then....-2>-3
If d=-100 & e=-200 then ......-100>-199

So here d is not greater than 1.