I generally use FGDBs, but it really depends on what you need to store. Like the other posts have mentioned, Personal GDBs have are outdated and have size issues.

Using a geodatabase allows you to set up a topology, which you can't do with plain shapefiles.

Shapefiles hold the exact coordinates of the shapes. In a GDB, the locations are snapped to the nearest point on a grid. I've not run into problems with this, it can cause small changes in data. Also, Shapefiles are easier to access with 3rd party tools. ESRI has an API for the FGDB, but it's new compared to existing ways to interact with Shapefiles.

I generally use FGDBs, but it really depends on what you need to store. Like the other posts have mentioned, Personal GDBs are outdated and have size issues.

Using a geodatabase allows you to set up a topology, which you can't do with plain shapefiles.

Shapefiles hold the exact coordinates of the shapes. In a GDB, the locations are snapped to the nearest point on a grid. I've not run into problems with this, it can cause small changes in data. Also, Shapefiles are easier to access with 3rd party tools. ESRI has an API for the FGDB, but it's new compared to existing ways to interact with Shapefiles.