Realism is a position that principles (like justice) and entities (like cats): 1. exist in reality, and; 2. are knowable by the intellect (Realism is basically articulated common sense).
The difference between the ‘Extreme Realism’ of Plato and ‘Moderate Realism’ of Aristotle is in the entities (things, especially living things, like cats and eggs and people).
Plato thinks that principles like justice are: i. universal, and; ii. transcendent (i. universal, because it’s the same one principle behind many concrete just ‘things,’ like just people or just courts of law, yet; ii. transcendent, since such principles doesn’t intrinsically belong to who they are; for example, ‘just’ people can turn in to ‘evil’ people – the just-ness transcends their humanity).
Because of that, Plato thinks that transcendence (ii.) must be the case with the essence or nature of entities too. However, Aristotle thinks the more common sensical thing and says that, even though their nature is universal (i.) within a species, entities have their natures within them (not ii. – cats all possess the nature of a ‘cat’).
So, if the Platonic focus is more transcendent and other-worldly. Aristotle would bring in perhaps a more pervading sense of highlighting transcendence and nature of things from within the concrete. I think good Realist philosophy upholds both in their respective areas, but, either way, admitting that such principles and entities exist (and are knowable) is a very good start in authentic philosophy.