Do SQL service packs effect SQL Compatibility mode?
Most likely the application will work, however, generally when a vendor has this statement, they have not tested their application. They have not certified the application will still work once the SP or CU or version has been applied. Thusly, if you go to an "unsupported" version, they will not support you on any issues. 

I would not rely on a vendor's website documentation as the end all statement for supportability. Sometimes, they have not updated their documentation yet. Give them a call to determine what the true support options are, and then get that in writing.