 Solution

Use the characteristic function to establish convergence in distribution by the Lévy continuity theorem, especially when exponential moments do not exist; a Cauchy distribution, for example, has a characteristic function but no finite moment-generating function on a neighborhood of zero.

Use the moment-generating function for an exponential-tail estimate: for $t>0$ with $M_X(t)<\infty$, Markov inequality givesOptimizing over $t$ produces a Chernoff bound. The modulus-one characteristic function does not directly give this positive exponential bound.