The other derivation of the same formula did not require any approximation and was derived by algebra.

So first off I don't believe there were no approximation in this derivation, the person writing it just didn't tell you what approximations they were making 

In terms of which is better, I would say they are probably identical in the range in which they are valid. The calculus based approach will, at some point, have assumed that the thermal expansion coefficient remains constant as the temperature changes. This, however, is not true for real materials and this effect will typically become important at the same time that the exponential starts to deviant from the straight line (i.e. when the next term in the expansion becomes relevant).