Both Connections and Comprehensive Google Map Plugin require the use of the Google Maps API. When this API is included on the page twice, it can cause the mapping functions of Connections and its templates to fail.
Connections does come with an option to disable including the Google Maps API. This option can be found on the Connections : Settings admin page under the Advanced tab. Unchecking this option might resolve the issue in some cases. If the Google Maps API is not loaded by Comprehensive Google Map Plugin before Connections requires it, the mapping functions will continue to fail to work in Connections. This is because of the load order of the plugins which is controlled by WordPress and not Connections or Comprehensive Google Map Plugin.
The only other option is to use another plugin. A user suggested Leaflet Maps Marker as an alternative to Comprehensive Google Map Plugin.
NOTE: This FAQ was written based on Comprehensive Google Map Plugin 7.0.31