Akris is Swiss, family-run out of St. Gallen since 1922, and has built its reputation on taking things away: clean seams, few visible fastenings, and cloth chosen so the garment holds its line without much help from the inside. Double-face fabric does a lot of that work, since a cloth finished on both faces needs no lining to look finished.
Max Mara came out of Reggio Emilia in 1951 into a tradition where the coat is the main event rather than a layer over the outfit. Its outerwear generally carries more internal structure, and the silhouette reads as engineered rather than draped.
The cloth is where the two houses separate most clearly. Max Mara's name rests on camel hair and wool, most famously the 101801, the camel coat Anne-Marie Beretta designed for the house in 1981 and that it has produced ever since.
Akris works closer to the technical end of the wool trade, with double-face wools and cashmeres and the occasional synthetic used as a deliberate choice rather than a saving. Neither house prints micron counts on a garment, and no resale listing has access to them, so treat any such figure you see quoted in a description as decoration.
Read the shoulder first. Akris jackets tend toward a narrow, undemonstrative shoulder, with interest carried by the cut of the panels instead of by hardware or applied detail.
Max Mara coats more often use a fuller cut, a longer line, and a belt or wrap closure. That is why they hang well in a photograph and why they still work pulled over a jacket, which matters when you are buying blind and cannot try the layering.
Akris is the more expensive brand new, often by a wide margin, and frequently the cheaper brand secondhand. The reason is recognition rather than quality: a Max Mara coat is known well enough that a seller can name a price and get it, while an Akris blazer with the label tucked away inside goes out for a fraction of what the cloth and the cutting cost.
That gap is the whole opportunity, and it is widest on the plainest pieces, the ones with no obvious signature to search for. It closes fast on anything a seller can identify by name.