I am most familiar with the last century of the existence of the Papal States, so my answer largely applies to the 1800's. Also, I have not done a comparative study of neighboring contemporary states, so my knowledge is limited.
However, everything I have read points to the Papal States being a fairly mundane minor regional power. Economically it was never able to be a powerhouse, and most of the time it lagged behind other regional powers and lagged far behind the larger continental powers. Temporally, the pope was a monarch like many others. His administration was largely made up of clerics (laypersons were integrated most fully by the last pontiff of the Papal States: Pius IX). Railroads were introduced fairly late, as pontiffs prior to Pius IX opposed their construction. The Papal Military was rarely able to exert itself outside its own territory and was usually little more than a minor deterrent to large armies.
If you read a general history of the papacy such as A History of the Popes: From Peter to the Present by John W. O'Malley S.J. you'll find a history of riots, plagues, military defeats, corruption, and all of the other mundane temporal challenges faced by other countries throughout history. Some popes were able to stock their treasury, others wasted their funds, and a few spent their treasury wisely. With few exceptions the Papal States was limited economically and militarily. The prestige of the papacy sometimes protected the nation from outside interference, but it also made it a target. Basically, the day-to-day life of the Papal States was much like any other minor nation throughout its history.
There is a history of how the Papal States interacted with its Jewish residents. I am not an expert in this area. The Popes Against the Jews: The Vatican's Role in the Rise of Modern Anti-Semitism by David Kertzer has an excellent background on how the Catholic Church contributed to anti-Semitism during its history. However, I find his argument to weaken as it reaches the 19th and 20th century. This critique of the book by Ronald Rychlak covers most of the criticisms I have of Kertzer's treatment of the 20th century papacy. Kertzer's work on earlier actions is better, but still misses some aspects of how Catholic teaching on this issue evolved.
Perhaps another of our experts can fill in the many gaps in my answer, but followup questions are always encouraged.