Why do we use dG < 0 to describe a spontaneous process?
Maybe its best to see how we generally derive the Gibbs energy... If you think about this then the answer should be clear ;) 

The Gibbs potential is a function of $P$, $T$ and $N$. That is to say:
\begin{equation}
G=G(T,P,N)
\end{equation}
Where $T$ is the temperature, $P$ is the pressure and $N$ is the number of particles.
\begin{equation}
dG=\frac{\partial G}{\partial P}\bigg|_{T,N}dP+\frac{\partial P}{\partial T}\bigg|_{P,N}dT+\sum _{i=1}^{I}\frac{\partial G}{\partial N_i}\bigg|_{P,T,N_j\neq i}dN_i
\end{equation}
Use some Maxwell relations (Look these up):
\begin{equation}
dG=VdP-SdT+\sum ^{I}_{i=1}\mu _i dN_i
\end{equation}
By definition we state the following formula at constant temperature and pressure:
\begin{equation}
dG=\sum _{i=1}^{I}\mu _idN_i
\end{equation}
Therefore you are actually just expressing the Gibbs energy in a different way when your book quotes the chemical potential. In fact the very definition of the chemical potential is:

\begin{equation}
\mu _i=\bigg(\frac{\partial G}{\partial N_i}\bigg)_{P,T,N_{j\neq i}}
\end{equation}

I don't know your mathematical ability but if it helps...its pretty clear that this is just a simple Legendre transform of the internal energy to switch between $U$ and $G$. 
(As an aside), with a little product ruling we get:
\begin{equation}
dG=\sum _{i=1}^{I}\big(\mu _idN_i+N_id\mu _i\big)
\end{equation}
This leaves us with the Gibbs-Duhem relation:
\begin{equation}
\sum ^{I}_{i=1}N_id\mu_i=-SdT+VdP
\end{equation} 
Which is a handy result. 

Also just to add on the end here about nucleation points... lets imagine a precipitation in solution....As the size of the nucleus grows (as more bits "clump" together) the Gibbs energy of the particle will rise ... this is due to two competing effects:

1) the interface interaction (increasing energy) 

2) the volume free energy (decreasing energy)

The total energy is just the resultant of these processes ... at small size the surface interactions outcompete the volume energy liberation .... as the radius of the particle increases we see a switch.