Okay, here's my try at it. Mind you these are my personal observations, and may not reflect the intent of each of the successor states.
Capellan Confederation: Space North Korea/Communism/Han-Chinese. As the smallest and youngest of the successor states, the Confederation is a paranoid police-state, that spends as much time spying on and thought-policing its citizens. The citizens of the Confederation, and by extension their property, are property of the state, with their very lives more than once having been sacrificed to drive back invaders rather than to let them get a foothold in Capellan space. Pros of the Confederation is its strong unity and superb (if heavily controlled) education system. The cons are that your life is state property.
Draconis Combine: Space Imperial/Shogunate Japan, sprinkled with Spartan ethics and fascism. Settled on some of the most resource poor planets in the Inner Sphere, house Kurita fosters a spartan and martial outlook amongst its people. Whereas the other successor states formed to ensure their survival and independence, the Combine formed from a desire for conquest. Pros are that the Combine has some of the fiercest and most loyal troops. On the other hand, the strict adherence to Bushido makes many Combine soldiers more concerned about their personal glory and honor over what is good for the military as a whole, up to an including committing suicide rather than face the shame of defeat.
Federated Suns: Space America/England/France. Spreading freedom whether you asked for it or not. The ruling Davion dynasty are enamored with medievial imagery and culture, and grants its subjects the most freedoms of any of the great houses. Pros are it's got the best trained military and some of the most advanced technology in the Inner Sphere. Cons are that if you live in the outback, you're basically barely above 19th/20th century living standards, and schooling/literacy isn't a garauntee.
Free World League: Space EU. A confederation of several distinct, sovereign states, with their own cultures and customs, unified under a federal government. The League is governed by a three-way tug of war between the national government, the federal parliament, and the increasingly dictatorial Captain General of the Free Worlds Leage Military. The Captain General, often a scion of House Marik, has a number of emergency priviliges that can override the will of parliament granted by an in-perpetuity state of Crisis. Pros of living in the Leage is a semblance of democracy and independence. Cons are that you get to watch those same principles of democracy and independence slowly wither and die. Also, the tug of war between the nations that make up the league, parliament, and the Captain General ensures the League military is bogged down in bureaucracy and in the worst case civil war.
Lyran Commonwealth: Space Hansa/Holy Roman Empire (the German one). Founded by a triarchy of merchant houses, before being taken over by the charismatic Steiner family, the Commonwealth is fascinated with wealth, nobility and social standing. As a merchant empire, its citizens enjoy some of the best standard of living in the Inner Sphere. Alas, the high command of the military is plagued by Social Generals, officers chosen for their name and their connections rather than any skill or merit. They have some of the best military equipment in the inner sphere, but that advantage is bungled in the hands of these Social Generals. Pros of living in the Commonwealth is a good standard of living and a relatively free populace. The cons lie in the realm's fascination of rank and nobility, making advancing ones social standing difficult. Also, weath breeds corruption, and those who do not have the wealth suffer from this corruption the most.