There are different indexes because they were created by different governmental agencies at different times. We're lucky to have both because, as you say, each has strengths and weaknesses. They also serve as valuable cross-references when a card is missing or illegible. There are other kinds of CW pension indexes too but those two are the most commonly used.
Both indexes include the veteran's name/regiment and the application date, Application number, and Certificate number for each person who applied with reference to a soldier's service. Then each index has some information that the other doesn't.
The General Index (Microfilm publication T288) can be found on Ancestry.com and FamilySearch.org, among other places. In the case of a widow's pension, only the General Index lists her name. It also shows the state where each applicant was living. Most importantly, only the General Index shows a C or XC number to reflect when a pension was still active after about 1934. If there is such a number, it is critical to finding the actual file. While most pensions are stored in Washington D.C., most C/XC pensions are stored in St. Louis, MO.
The Organizational Index (T289) can be found on Fold3.com. Unlike the General Index, it includes fields for the soldier's rank, terms of service, and date/place of death. These types of data aren't always present but it sure is nice when you find them there!
As the names suggest, the indexes are sorted in different ways. If you want to see the pension card for everyone in a specific company or regiment, etc., then it is easiest to scroll through the Organizational Index whereas the General Index is sorted by the state and name of the veteran. When I'm looking for a Civil War pension, I always consult both.
There are other differences, of course. The Organizational Index does not include Navy pensions, for instance. I've written a 7-part series of blog posts (!) about Civil War pension indexes and how to work with them but I'm still finishing off a few details. I'll post a link here when it is ready.