Google Maps API

Global reach with data from almost everywhere.
Consistent formatting for street addresses.
Consistent and predictable administrative areas; cities are (almost) always cities (and not regions, for example).
Map translations are available, meaning you can set the language of the map.
Custom Styling with the Custom Maps addon.
Users cannot fix errors, but incorrect data can be reported to Google.
Can be costly once free quota has been exceeded.

Open Street Maps API

Free.
Several map styles (tilesets) available with different styles through the Custom Maps Addon.
Data errors can be correct by directly editing the map on the OSM site.
Global community, but a much lower overall volume of data.
Inconsistencies in address formatting, especially between countries.