Check this answer on SO too.
You need to update the config files of the desktop/web clients as follows.
<configuration>
<runtime>
<assemblyBinding xmlns="urn:schemas-microsoft-com:asm.v1">
<dependentAssembly>
<publisherPolicy apply="no" />
<assemblyIdentity name="Aspose.Words" publicKeyToken="716fcc553a201e56" />
<bindingRedirect oldVersion="11.0.0.0-13.6.0.0"
newVersion="13.7.0.0"/>
</dependentAssembly>
</assemblyBinding>
</runtime>
</configuration>
Please also note that there might be breaking changes in API which might lead to other errors. For example a method or property which existed in 11.0 is deleted in 13.7. The exception will be thrown when the client application calls the specific method/property. So, you must verify in development environment that the new version will not break your application. If it works, then you can just replace the old DLL with new version.
The config file is loaded at runtime, so you can update the config files where your applications are deployed.
I work for Aspose as a Developer Evangelist.