The gtk_widget_activate_action() function is a C convenience wrapper around gtk_widget_activate_action_variant() that does the g_variant_new() dance for you.
The gtk_widget_activate_action_variant() function will either call g_action_group_activate_action() on the widget, or will traverse the widget hierarchy until it hits a top level, and call g_action_group_activate_action() there. So, up to a point, GtkWidget.activate_action_variant() is mostly a wrapper around the interface method GActionGroup.activate_action().
The overloading of the method name is kind of unfortunate, but in general, both methods will end up calling the same implementation.