Checked the error log. We had duplicated an existing context (the sites will be virtually the same) and discovered that some resources were missing aliases, probably because the aliases in the original contexts took from the name of the resource and had some slashes (i.e. /) in them.
We reset the aliases to proper aliases and when we visit the second context's URL we no longer get errors like this in the log:
[2013-11-13 14:08:52] (ERROR @ /connectors/resource/index.php) Resource URI .html already exists for resource id = 64; skipping duplicate resource URI for resource id = 74
However, we are getting this error:
[2013-11-13 14:12:05] (ERROR @ /connectors/resource/index.php) [ContextRouter] You may have conflicting http_host definitions. Overwriting context web with web for host default
In the System Settings for ContextRouter we have these:
contextrouter.defaultContext: web
contextrouter.includeWww: Yes (also tried No).
Don't see where there is an http_host definition conflict.