An answer to your second question: Stoll constructed many 2-step nilpotent groups such that there exist generating sets $S$ and $S'$ such that the growth series with respect to $S$ is rational and with respect to $S'$ is transcendental. See

M. Stoll, Rational and transcendental growth series for the higher Heisenberg groups, Invent. Math. 126 (1) (1996), 85–109.

As for your first question, I don't know if it is intrinsically interesting, and I am not aware of any applications of these results. However, what is interesting are the vast array of tools and techniques that are used to study it. It is a good proving ground for many ideas in geometric group theory (especially those related to things like regular languages). There is an enormous literature on this topic. The introduction to my paper

A. Putman, The rationality of sol manifolds J. Algebra 304 (1) (2006) 190-215.

summarizes most of the papers concerning it that I am aware of. It can be downloaded from my webpage here. The only ones I know about that came out after it are

1. My student Corey Bregman's paper "Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles", available here.

2. Duchin-Shapiro's paper "Rational growth in the Heisenberg group", available here.

An answer to your second question: Stoll constructed many 2-step nilpotent groups such that there exist generating sets $S$ and $S'$ such that the growth series with respect to $S$ is rational and with respect to $S'$ is transcendental. See

M. Stoll, Rational and transcendental growth series for the higher Heisenberg groups, Invent. Math. 126 (1) (1996), 85–109.

As for your first question, I don't know if it is intrinsically interesting, and I am not aware of any applications of these results. However, what is interesting are the vast array of tools and techniques that are used to study it. It is a good proving ground for many ideas in geometric group theory (especially those related to things like regular languages). There is an enormous literature on this topic. The introduction to my paper

A. Putman, The rationality of sol manifolds J. Algebra 304 (1) (2006) 190-215.

summarizes most of the papers concerning it that I am aware of. It can be downloaded from my webpage here. The only ones I know about that came out after it are

1. My student Corey Bregman's paper "Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles", available here.

2. Duchin-Shapiro's paper "Rational growth in the Heisenberg group", available here.

An answer to your second question: Stoll constructed many 2-step nilpotent groups such that there exist generating sets $S$ and $S'$ such that the growth series with respect to $S$ is rational and with respect to $S'$ is transcendental. See

M. Stoll, Rational and transcendental growth series for the higher Heisenberg groups, Invent. Math. 126 (1) (1996), 85–109.

There is an enormous literature on this topic. The introduction to my paper

A. Putman, The rationality of sol manifolds J. Algebra 304 (1) (2006) 190-215.

summarizes most of the papers concerning it that I am aware of. It can be downloaded from my webpage here. The only ones I know about that came out after it are

1. My student Corey Bregman's paper "Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles", available here.

2. Duchin-Shapiro's paper "Rational growth in the Heisenberg group", available here.

An answer to your second question: Stoll constructed many 2-step nilpotent groups such that there exist generating sets $S$ and $S'$ such that the growth series with respect to $S$ is rational and with respect to $S'$ is transcendental. See

M. Stoll, Rational and transcendental growth series for the higher Heisenberg groups, Invent. Math. 126 (1) (1996), 85–109.

As for your first question, I don't know if it is intrinsically interesting, and I am not aware of any applications of these results. However, what is interesting are the vast array of tools and techniques that are used to study it. It is a good proving ground for many ideas in geometric group theory (especially those related to things like regular languages). There is an enormous literature on this topic. The introduction to my paper

A. Putman, The rationality of sol manifolds J. Algebra 304 (1) (2006) 190-215.

summarizes most of the papers concerning it that I am aware of. It can be downloaded from my webpage here. The only ones I know about that came out after it are

1. My student Corey Bregman's paper "Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles", available here.

2. Duchin-Shapiro's paper "Rational growth in the Heisenberg group", available here.