Thermodynamics needs differentials. However, it is also the field where we can witness the maximum departure from the usual mathematical definitions. It is probably a good idea to try a systematic summary of mathematical definitions before discussing folklore and mathematical facts about differentials in thermodynamics.
Mathematical differentials in a nutshell
Given a real function of $N$ real variables ($f(x_1,x_2,\dots,x_N$), we can define the differential $df$, at a point $\bar x=(\bar x_1,\bar x_2,\dots,\bar x_N)$ as
$$
df= \sum_{i=1}^N \left[\frac{\partial{f}}{\partial{x_i}}(\bar x)\right] (x_i - \bar x_i).
$$
From the definition is immediate to see that we can write $dx_i = x_i- \bar x_i$, and then
$$
df= \sum_{i=1}^N \left[\frac{\partial{f}}{\partial{x_i}}(\bar x)\right] dx_i.
$$
It turns out that, at the point $\bar x$, $df$ is the best linear approximation of the variation $\Delta f = f(x)-f(\bar x)$, in the sense that
$$
\Delta f - df = \mathcal{O}(dx_1^2,dx_2^2,\dots,dx_N^2).
$$
Moreover, from the definition, it is clear that $df$ is a linear function of the variables $(x_1,x_2,\dots,x_N)$, and in general a non-linear function of the variables $(\bar x_1,\bar x_2,\dots,\bar x_N)$.
After introducing the differential of a function, we can introduce a differential form $dy$, i.e. a function of $(x_1,x_2,\dots,x_N,\bar x_1,\bar x_2,\dots,\bar x_N)$ defined as
$$
dy = \sum_{i=1}^N A_i(\bar x) dx_i,
$$
where, in general the functions $A_i(\bar x)$ are not the partial derivatives of a function of $N$ variables.
A differential form $dy$ is called exact if there is a function $y(\bar x)$ such that
$$
A_i(\bar x) = \left[\frac{\partial{y}}{\partial{x_i}}(\bar x)\right]. \tag{1}
$$
Finally, we can define a line integral of the differential form $dy$ over the curve $\gamma$, whose parametric expression is
$$
\bar x(t)= (\bar x_1(t),\bar x_2(t),\dots,\bar x_N(t))~~~~~t \in [a,b],
$$
as
$$
\int_{\gamma}dy =\int_a^b dt \sum_{i=1}^N A_i(\bar x(t)) \frac{d \bar x_i}{dt}. \tag{2}
$$
It turns out that if the differential $dy$ is exact, $ \int_{\gamma}dy = y(\bar x(b)) - y(\bar x(a))$.
Notice that all the previous definitions and formulae could be generalized to the case of differential forms on a differential manifold. However, such a generalization is a kind of overshooting in Thermodynamics, where the space of the states is more naturally described as a cone in $\mathbb R^N$.
How differentials are used in Thermodynamics?
All previous definitions and formulas have a direct application in Thermodynamics. In particular, the line integrals of differential forms (exact or not) are defined as in formula $(2)$, and this should answer part of your question.
There is a unique point where some mathematical confusion is introduced in many textbooks on Thermodynamics. It is the introduction of the so-called inexact differentials.
Strictly speaking, a non-exact differential form, i.e., a differential form with no function $y$ satisfying condition $(1)$, could be called inexact. However, this is not how some people introduce the inexact differentials.
An example is the case of the internal energy as summarized by the first principle of Thermodynamics:
$$
\Delta U = U(B)-U(A)=Q + W
$$
where $W$ and $Q$ are the total work, and the total heat exchanged in a thermodynamic transformation from an equilibrium state $A$ to another equilibrium state $B$.
The internal energy is a function of the state, and for a quasi-static reversible transformation from a state to a near state, we can approximate a small variation $\Delta U$ by the differential $dU$.
It is attempting at this point to write for a completely general process between such neighbor states.
$$
dU = dQ + dW.
$$
However, $Q$ and $W$ are not functions of the state. Therefore the introduction of a special symbol $\delta$ or $-\hspace{-1ex}d$ to make explicit that they are not differentials of a function of state:
$$
dU = -\hspace{-1ex}d Q + -\hspace{-1ex}dW.
$$
Unfortunately, such a notation hides the fact that in the case of a non-quasistatic process, not only $Q$ and $W$ are not differentials of a function, but they are neither functions of the state variables.
Therefore, I would leave the term inexact differential only to indicate a real differential form of the state variables whose integral is path-dependent. For instance $- \int p(T,V,N)dV$ (defined only for a quasi-static process).
