Chaotics believe in the individual's right to decide and that combining into groups usually requires too many compromises to allow a creature to achieve the best in life (whatever that may be, guided by whether they are Good or Evil). So chaotic societies are generally going to be broken into small groups which share a lot of the same culture and goals, so there is little compromise needed.

Chaotic Evil societies are likely to be dominated by the strong, with the weak either cowed or tagging along to pick up the scraps left by the leaders. There is no conception of not "rocking the boat" for the good of all. If a leader is unable to dominate, s/he will be hung out to dry by the next strongest.

Only under extreme threats will large numbers band together for mutual support against some common enemy; the rest of the time other Chaotic Evil creatures will be the enemy as often as Lawful Good will be.

Chaotic Good societies may elect leaders for special needs - for example, forming a posse to hunt down some orcs, or picking a skilled diplomat to meet outsiders. Such leaders will have strictly temporary positions and will only be chosen if they can represent enough individuals' interests. Otherwise, those individuals are likely to request that they be picked. This can lead to squabbling, of course, but less so than with Evil creatures.

Alignment is not a reflection of reality, it is a representation of a fantasy reality where evil and good can actually physically affect the world - where being a bad person can result in your children being deformed or being a good person can result in straight teeth and hair that always looks nice even after falling in a swamp.