map projections that minimises differences
I am sorry, but this type of linear map can't exist. You said that you want to minimize differences, so you want to make small countries bigger. But unfortunately, big countries become bigger too (because of linear transformation), so that's not the solution.
I can only think of scaling individual portions of some standard projection. For example, you could scale Europe a little bit because of many small countries. You could also just scale the islands in the Oceania, but not the distances between them. Oops, it seems like we got too far into cheating, but as long as scaling countries to minimize differences is the only thing you want to do, there is no problem. Of course, your decision is the last.
As @KeithMcClary pointed out in his comment, this type of map is called cartogram. Thank you for this useful piece of information!