If there is one question I've been asked more than any other over the years, it's whether or not a church management system should include accounting or not.
In short: No, but it's ok if it does.
Let me explain: Church Management Systems need to include contribution data, because that's the best way to keep donations associated with the proper people without duplication of data. In fact, properly attributing contributions and creating year end reports and giving letters is one of the single most important functions of a church database. At a prior church I wrote a database specifically around this need after the IRS regulations changed in 1994, so I understand this area very well.
However, after those contributions are recorded, the actual entry into the finance system is a simple one. Generally a single journal entry that matches the deposit and assigns the income to the necessary accounts is all that is required. This takes, at most, a few minutes a week. So, while it's ok for a church management system and a finance system to be together in one package, it is completely unnecessary. What is important is that you have a church finance system that works the way you want it to, and supports your organization with accurate financials and reporting, and that you have a church membership database that meets your needs and helps your church minister more effectively to the people God has entrusted to your care.
It is not worth compromising either function for the sake of integration. Buy the best financial package to meet your needs, and buy the best church management solution to meet your needs.