Laura,
In linear models without confounding or interaction, the difference method= product method = Natural direct or indirect effects. And also the natural direct effect=controlled direct effect.
So you can think of the difference method as the starting point. Now you have some problems that might come up, and all of the other tools are ways to help you deal with that. The first problem is non-linear models. If for example, it's a binary outcome and you have odds ratios, can you just take the difference? No. But you can use the formulas in Tyler's paper to account for the non-linearity and the intuition is the same.
Another problem is if the mediator modifies the direct pathway. Now there is more than one "direct" effect. You can ignore than and stick with the natural direct effect or you might want to understand how the direct effect would differ, depending on the value the mediator takes. Again, you can use the counterfactuals to state very precisely what you're estimating and the formulas in Tyler's paper to estimate it.
Another problem is if there is a confounder of the mediator and the outcome which, for whatever reason, you don't want to just control for in your model. Knowing these new tools let's you use methods like inverse probability weighting to handle such a confounder.
So, if you get the difference method, you are 80% of the way there intuitively, but realize it has limitations and if you move out of very simple situation, you need a better tool.
Maria
