Why does marginal cost (derivative of total cost) differ from variable cost at each level?
For $Q_1 = 1, Q_2 =2$, you calculate

$$TC(Q_2) - TC(Q_1)=7$$
which is the additional (incremental) cost to produce the second unit of output, and then you compare this to

$$\frac {dTC}{dQ} |_{Q=Q_2} = 12$$

which is marginal cost when you are already at production level $Q_2$. From this observation alone, the two are not comparable.  

As for practical value: assume marginal revenue is a straight line at $P=8$. What to do? The "marginal cost equals marginal revenue" leads to  $3Q^2 \leq 8 \implies Q^*= \sqrt {8/3} \approx 1.633$. What's the problem with that?  

Well, the product we are examining may not be divisible so much as to make incremental cost adequately close to marginal cost. This not-sufficient divisibility  may force us to resort to integer-value calculus, and use the concept of incremental cost instead : produce the highest quantity for which incremental cost is lower or equal to incremental revenue - and we will obtain the correct answer $Q^* =2$. 

Note: "sufficient divisibility" is not a strictly "physical constraint" but is judged also on economic considerations. If we are contemplating producing, say, cars to the range of hundreds of thousands, then, even if each car costs say 20 thousand euros, the pains and computational inconvenience of integer-valued calculus may not worth the small reduction in profits if (by using marginal calculus) we end up producing one unit more than optimal given indivisibility.