My education in SDEs is all self-taught, so please excuse any inaccuracies below. In particular, my memory is fuzzy on issues of SDE vs associated PDE, and backwards versus forwards equation, for instance.
In differential equations, and in applications, we look at differentials often because that's all that we can infer in a straightforward way. Locally, we are able to make arguments about how something should behave, and we integrate to try to gather what is going on over global distances (finite, rather than infinitesimal).
The solution may depend too much on what else is going on. Initial conditions, boundary conditions. The local differential equation may be true in general, but the actual global relative values may depend greatly on the larger environment. For instance, Maxwell's equations, even in simple distributions, locally will give the same differential relation between charge and electromagnetic field, but the actual solutions of this depend greatly on the charge distribution.
In Black-Scholes-Merton, the non-arbitrage behavior is locally given by the same SDE, but the solution may vary greatly depending on for instance what sort of stock option behavior is given for the final time boundary condition. Also, the chain rule is important in SDEs, and that should happen at the differential level.
Also, and my knowledge of this very informal, I don't consider the $dW_t$ term to be so imprecise. It is probably often taught imprecisely, and with quite a bit of hand waving, but it is not hard to come up with convincing arguments with regards to its behavior. I have in my notes things like $dW_t=W_t\sqrt{dt}$, $dW_t=\lim_{n\to\infty}\sum_{i=1}^nY(ih)\sqrt{\frac{t}{n}}$, where $Y(ih)$ is $-1$ or $1$ with equal probability $p=1/2$ (this is more or less from Finan's text on MFE, a free online textbook). I am pretty sure I didnt' write that right, but don't want to check details right now, so hopefully someone more motivated can check. Also, I don't think that sort of representation is unique, but that it suffices to use that one will work. This reminds me of how you can represent the delta function as the limit of various distributions.