What is the specific difference between first and second kind of Christoffels? Though mathematically, the Christoffels symbols of the first and second kind are different because of the presence and absence of given metric in the given basis. How could we understand this state in terms of geometric view in case of the spherical coordinate system?
 A: Before understanding the difference between Christoffel symbols of the
first and the second kind, it is necessary to understand the difference
between the basis vectors $\vec{e}_i$ and the dual basis vectors $\vec{e}^i$.
Consider curvilinear and non-orthogonal coordinates
($x^1$, $x^2$, ...). Note that by convention coordinates
always have an upper index.

In these curvilinear coordinates the basis vectors $\vec{e}_i$ (with
a lower index) are defined to be tangential to the coordinate lines.
These basis vectors are intuitive and important because they
directly appear in the decomposition of a position difference vector
$$d\vec{x}=\vec{e}_i\ dx^i \tag{1}$$
On the other hand, the dual basis vectors $\vec{e}^i$ (with an upper
index) are defined by $\vec{e}^i\cdot\vec{e}_j=\delta^i_j$.

That means the dual basis vectors are perpendicular to the coordinate lines.
Hence these dual basis vectors are of less practical importance.
There is no useful decomposition similar to (1) like
$$d\vec{x}=\vec{e}^i\ dx_i$$
because coordinates $x_i$ with a lower index just don't make sense.
Because the coordinates are curvilinear, the basis vectors $\vec{e}_i$
vary from point to point.
Their changes can be decomposed with respect to the coordinate
changes ($dx^j$) and the basis ($\vec{e}_k$).
Written with differentials we have
$$d\vec{e}_i=\Gamma^k{}_{ij}\ \vec{e}_k\ dx^j  \tag{2a}$$
or equivalently (written as partial derivatives)
$$\frac{\partial\vec{e}_i}{\partial x^j}=\Gamma^k{}_{ij}\ \vec{e}_k. \tag{2b}$$
The coefficients appearing here are the Christoffel symbols
of the second kind.
Thus this decomposition can be considered as the definition for these Christoffel
symbols.
On the other hand, the change of base vectors can also be decomposed
with respect to the coordinate changes and the dual basis
$$d\vec{e}_i=\Gamma_{kij}\ \vec{e}^k\ dx^j \tag{3a}$$
or equivalently
$$\frac{\partial\vec{e}_i}{\partial x^j}=\Gamma_{kij}\ \vec{e}^k \tag{3b}$$
The coefficients appearing here are the Christoffel symbols
of the first kind.
This equation is less useful because the dual basis
vectors $\vec{e}^k$ are less useful (as explained above).
