The GAI is a composite of the VCI and PRI subtests. The FSIQ is a composite of VCI, PRI, WMI and PSI. The GAI is usually higher in gifted kids because while they may have advanced verbal (VCI) and analytical (PRI) capabilities, their working memory (WMI) and processing speed (PSI) scores may be closer to average.
When people refer to IQ, they usually mean the FSIQ. But really, there is no "actual" IQ associated with a person... any of these measures is only a snapshot of a small subset of someone's abilities. People will get different scores on different tests, and different scores on different days.
I guess I would say that the GAI and the FSIQ can be more or less relevant depending on the situation.
You can read more about the GAI here:
WISC-IV Technical Report #4: General Ability Index