George Pólya writes that a method is a trick you can use more than once.

(Although I think he ascribes this to professors who are not good teachers.)

I would classify a trick as something that happens to work but isn't rigorous. Like treating dy/dx as a fraction sometimes works, but only under certain conditions.

In my mind, a trick is something that applies to unusual and specialised cases, whereas a method is something that can apply to a broad, well-defined class of problems.

What's your favourite example of where it doesn't work? Physics is full of quasi-infinistesimal quantities and I always like good counter examples (ideally without invoking something like the blamange function or similar....)

It basically works without issue in 1d, simply because dx and dy can be considered modular forms and dy is exactly dx times the derivative dy/dx. You can even put an integral sign in front of them and calculate the corresponding integral.

Where this doesn't work is if you have more than 1 dimension. Then you need to deal with the added complexity of integrating modular forms and the fact that in 2D you don't have df = (df/dx) dx but df = (df/dx) dx + (df/dy) dx. The chain rule also changes into a matrix product, rather than a simple dz/dx = dy/dx dz/dy.

Somewhat silly example: given a plane in 3 dimensions defined by x+y+z=1, we have ∂x/∂y * ∂y/∂z * ∂z/∂x = -1.

Off the top aren't there differential equations that are inseparable? ie you cannot just pretend x and y are x(t) and y(t).

No, a trick is a shortcut with respect to the more tedious method. By definition anything that works also formally works, otherwise it wouldn't.... work.