$f=f(x_1,…,x_n)$ and $g=g(x_1,…,x_n)$ are two differentiable functions.
$\frac{\partial f}{\partial x_1} = \frac{df}{dx_1}\big|_{x_2,…x_n}$ is the partial derivation with respect to $x_1$ while leaving the other variables as constants.
What is $ \frac{df}{dx_1}\big|_{g,x_3,…x_n}$ e.g. the infinitesimal change of f variating $x_1$, while leaving $x_3,…,x_n$ and g constant?? In the first case $x_1$ variates over the $x_1$-axis, in the second case $x_1$ variates over some curve where g is constant (this is a curve, cause we have the additional restriction that $x_3,…,x_n$ is constant too). Is this the part of the space, where g is equal to some constant a manifold?
$\partial f/\partial x_1=df/dx_1|_{x_2,\dots,x_n}$ is just the definition of partial derivative of a function of several variables. When you fix $x_2,\dots,x_n$, $f$ becomes a function of single variable $x_1$ and therefore the partial derivative and usual derivative coincides.
When $g$ is held constant, $f$ is restricted to a hypersurface (dimension $n-1$) in $\mathbb R^n$. But the easier way to understand the partial derivative $df/dx_1|_{g,x_3,\dots,x_n}$ is w.r.t. to a new function: $$ \hat f=\hat f(x_1,g,x_3,\dots,x_n) $$ If we denote the coordinate transformation by $\Phi$ (which should be a local diffeomorphism): $$ \Phi:(x_1,\dots,x_n)\to(x_1,g(x_1,\dots,x_n),x_3,\dots,x_n) $$ $\hat f$ is then given by: $$ \hat f=f\circ\Phi^{-1} $$ So you will have to make sure the Jacobian: $$ d\Phi=\left[ \begin{array}{ccccc} 1 & 0 & \cdots & & 0\\ \partial g/\partial x_1 & \partial g/\partial x_2 & \cdots & \cdots & \partial g/\partial x_n\\ 0 &0 & \ddots& & 0\\ \vdots &\vdots& \ddots & 1 & \vdots\\ 0 & \cdots & \cdots & 0 & 1 \end{array} \right] $$ is non-singular, which is equivalent to $\partial g/\partial x_2\neq 0$.