SSRS reachable by FQDN on local computer, but not from a different computer

I am having trouble getting my SSRS server to successfully cohabitate with IIS, after having to move the server to a different subnet. I have 2 static IPs for that server: .242 is defined explicitly in IIS bindings, and .161 is for SSRS. I have my SSRS Web Service URL set to "All Assigned", and the Report Manager URL set to (I am removing the http:// & https:// prefixes to get this posted) appreports.domain.com:80/reports & +:443/reports. DNS is pointing to .161. Also, SSRS points to an external database server.

What I am seeing is strange to me. I can browse to appreports.domain.com:80/reports on that server, and it looks great - I am at the SSRS Report Manager page, and all is well. When I call appreports.domain.com:80/reports from a different computer, I get an IIS error:

HTTP Error 404 - File or directory not found. Internet Information Services (IIS)

I am not sure why IIS intercepts the call when accessed remotely. In the System log, I see this:

A Kerberos Error Message was received: on logon session Client Time: Server Time: 17:10:4.0000 10/29/2013 Z Error Code: 0x7 KDC_ERR_S_PRINCIPAL_UNKNOWN Extended Error: 0xc0000035 KLIN(0) Client Realm: Client Name: Server Realm: DOMAIN.LOCAL Server Name: MSSQLSvc/VM-db.domain.local:1433 Target Name: MSSQLSvc/VM-db.domain.local:1433@Domain.LOCAL Error Text: File: 9 Line: f09

In the ReportServerService log, there are no errors - it is the picture of health.

I suspect there is an SPN issue, but I don't understand it well enough to comment. Thoughts most welcome!