In Calculus, the derivative w.r.t. $t$ of a single-variable function $f$ is denoted:
$$\frac{df}{dt}$$
While the derivative w.r.t. $t$ of a multi-variable function $f$ is denoted:
$$\frac{\partial f}{\partial t}$$
The justification being that the latter is distinct from the first in that it is a partial derivative.
Why does the latter warrant its own symbol? A single-variable function is just a special case of a function taking $n$ variables, so isn't the derivative of a single-variable function a partial derivative in a general sense?
The two operations seem to behave the same; is there any case where mixing them could be problematic?
In a single variable case, yes $\frac{df}{dx}=\frac{\partial f}{\partial x}$ but it's a bit odd to write this, because the whole point of a partial derivative is to emphasize that the $other$ variables are being treated as constants. Because you only deal with a single variable in Cal 1/2 (single variable calculus courses) it would seem pointless to mention partial derivatives. Also, the definition of the normal derivative is what you want in single variable calculus ,..... there will be times for both sorts of derivatives in multivariable calculus but only the normal derivative carries the same information that you learned from the single variable course.