Is $\Delta$ notation commonly used for the difference in a quantity between two objects? In a standard Atwood machine, the acceleration is
$$a = g\dfrac{m_1 - m_2}{m_1+m_2}.$$
Would writing this as
$$a = g \dfrac{\Delta m}{M}$$
where $M$ is the total mass be an abuse of notation to most physicists?
Alternatively, suppose that we are analyzing a heat engine and use $\Delta T$ for the difference in  temperature between the hot and cold reservoirs. Would this be clear notation, or confusing notation to most physicists?
In general, does $\Delta$ indicate only the change in a particular quantity between two times, or can it indicate the difference between two quantities at the same time?
 A: You can define $\Delta m$ or $\Delta T$ however you like as long as you DO specify what you mean by it. The usages you’ve suggested are perfectly natural to my eye.
That said, I’d strongly recommend against using this notation and just leaving it up to reader to figure out what it means. It is not THAT unambiguous.
Finally, I even if it were uncommon notation it would not be an abuse of notation. An abuse of notation is when some piece of notation is used in a slightly ambiguous or wrong way to Expedite reading or writing equations. You’re question makes it sound like you think “$\Delta$” is some special piece of notation or a function or something that takes a difference. This is not the case… $\Delta$ is just a Greek symbol. Sometimes we prepend it to other symbols to make a 2-letter variable like $\Delta m$, but this is just a single variable exactly like $x$ or $T$. It would be perfectly reasonable to define, for example, $\Delta= m_2 - m_1$ and in many cases something like this is actually done in practice.
And to be explicit, $\Delta$ IS used in practice to show the difference between two quantities of interest that have the same dimensions, not just the change in a single quantity over time or space or something. I personally don’t find your examples confusing or surprising in the least and my guess is that almost all physicists would be fine with it as well, but I can’t really speak for other physicists.
A: This use is extremely common in thermodynamics, at least. In the Clasius–Clapeyron equation, for instance, $\Delta$ is used to represent simultaneous differences between certain properties of two phases. It would be standard practice to use your $\Delta T$ example to calculate, say, the entropy generation associated with a thermally conducting rod connecting two heat reservoirs.
(As a side note, the notation in the Clausius–Clapeyron equation that most  confuses students is far from the use of $\Delta$—it’s the fact that $dP/dT$ is used to refer, without any clarifying annotation, to a coexistence curve rather than to the behavior of any single system.)
