Let $f:\mathbb{R}^n\to\mathbb{R}$ be a function. We can define the partial derivative of f with respect to its first argument using either a two-sided limit ($h\to 0$)
$$\lim_{h\to 0} \frac{f(x_1+h,x_2,\ldots,x_n)-f(x_1,x_2,\ldots,x_n)}{h}$$ or a one-sided limit ($h\to 0^+$)
$$\lim_{h\to 0^+} \frac{f(x_1+h,x_2,\ldots,x_n)-f(x_1,x_2,\ldots,x_n)}{h}$$
to see it as a directional derivative in the direction of the basis vector $e_1$.
Are there any peculiarities of one defintion vs the other? Do some interesting theroems/intuitions about differentiabiltiy change drastically?
You can consider the function $f(x)=|x|$ (of a single variable) and notice how the two definitions give different answers. Adding additional variables doesn't change anything.