Distribution for "joint multi-variate gaussian distribution" (joint MVG):
$$f_{X}(x) = \frac{1}{(2\pi)^{n/2}\prod \limits_{i=0}^{n}\sigma_i} ~~\text{exp}\bigg[-\frac{1}{2} \sum \limits_{i=1}^{n}\Big(\frac{x_i\mu_i}{\sigma_i}\Big)^2 \bigg]\tag{2}$$
Distribution for "multi-variate gaussian distribution" (MVG):
$$f_{\mathbf{x}}(x) = \frac{1}{(2\pi)^{n/2}~|\text{det }C|^{1/2} }~~\text{exp}\bigg[ -\frac{1}{2} (\mathbf{x}-\mathbf{\mu})^T C^{-1} (\mathbf{x}-\mu)\bigg]\tag{1}$$
Where:
$[n\times1] ~~~\mathbf{x} \Rightarrow \text{sample vector}$
$[n\times1]~~~\mathbf{\mu} \Rightarrow \text{mean vector}$
$[n\times n]~~~ C \Rightarrow \text{covariance matrix (symmetric positively defined matix)}$
What's the difference between the two?