It sounds like your network configuration has the wrong default gateway or netmask setting. With the same settings, can the module ping a server outside of the local network? If you try the HTTP client sample, can you connect to a remote web server?
Please post IP address, default gateway, network mask and DNS server settings when using DHCP versus using static configuration.
You could enable the IP_VERBOSE macro (or some of the Device Cloud related macros) in your program to see if you get debug messages that might help you determine the failure cause.