ping response "Request timed out." vs "Destination Host unreachable"

Destination Host Unreachable

This message indicates one of two problems: either the local system has no route to the desired destination, or a remote router reports that it has no route to the destination.

If the message is simply "Destination Host Unreachable," then there is no route from the local system, and the packets to be sent were never put on the wire.

If the message is "Reply From < IP address >: Destination Host Unreachable," then the routing problem occurred at a remote router, whose address is indicated by the "< IP address >" field.

Request Timed Out

This message indicates that no Echo Reply messages were received within the default time of 1 second. This can be due to many different causes; the most common include network congestion, failure of the ARP request, packet filtering, routing error, or a silent discard.

For more info Refer: http://technet.microsoft.com/en-us/library/cc940095.aspx

As I understand it, "request timeout" means the ICMP packet reached from one host to the other host but the reply could not reach the requesting host. There may be more packet loss or some physical issue. "destination host unreachable" means there is no proper route defined between two hosts.

Request timed out means that the local host did not receive a response from the destination host, but it was able to reach it. Destination host unreachable means that there was no valid route to the requested host.